This is an inherently portable project, so SMF batteries, power walls, and leaving on the work bench plugged in all the time really aren't good options for me. Need something light weight, small, and powerful.
I'm looking at an 11.1v LiPo 2200mAh that is described as being able to deliver 25C constant and 35C burst. Not really sure why it's capable of delivering so much more current than the 3.7v Li-Ion pack I was first considering which is only able to put out 6C burst. Oh, and it's half the cost.
The only converter that I'm seeing at the shop I go to that seems like it might work has a max input current of 7.5A / max output of 8A so I'm not sure if it will support this battery. My project won't be drawing more than 5A (burst) so I don't know if that will keep things under that maximums of converter.
Here are the listings:
Battery: https://abra-electronics.com/batteries-holders/batteries-polymer-lithium-ion/bat-25-3s-zippy-compact-2200-mah-3s-11.1v-25c-li-po-battery.html
Buck Converter: https://abra-electronics.com/voltage-regulator-modules/psm-150-dc-adjustable-step-up-converter-8-40v-to-12-60v.html
Charger: https://abra-electronics.com/batteries-holders/battery-chargers-testers/bat-charger5-7.4v-11.1v-lipo-2-3s-cell-balance-charger.html
Your thoughts?
Also, not really sure how to choose the right diodes. Here's the schottky diode with the highest current that's for sale at the local shop.
Says average forward current 3A. Not really sure what to do with this information.
AncientOracular:
Not really sure why it's capable of delivering so much more current than the 3.7v Li-Ion pack I was first considering which is only able to put out 6C burst. Oh, and it's half the cost.
Because the chemicals used inside of it are different. Li-ion is a generalized term for a wide variety of batteries.
The first battery pack you mentioned might have been a cobalt based cell whereas the new one might be a manganese based one.
It's half the cost because it's targeted at RC hobbyists unlike your first one which was targeted at electronics enthusiasts.
I'm not sure if it will support this battery. My project won't be drawing more than 5A (burst) so I don't know if that will keep things under that maximums of converter.
Yes it will since the current draw will be within the limits. The converter will only draw the current the project needs because it's a switched mode supply and not a linear converter.
Your thoughts?
Converter good
Battery good
Charger good - Just confirm with the seller whether the battery leads will fit into this charger.
However you mentioned that your project is portable, but the charger is supposed to be plugged into an AC power wall outlet ??? ?
Also, not really sure how to choose the right diodes. Here's the schottky diode with the highest current that's for sale at the local shop.
Nooo... I mentioned Schottky diode IF you were going with the 3.7V battery pack because the voltage drop of ordinary diodes would have brought down the input voltage below the boost converter's minimum requirement.
For this new battery setup you can use regular silicon diodes with average forward current rating of 6A or more.
Noobian:
However you mentioned that your project is portable, but the charger is supposed to be plugged into an AC power wall outlet ??? ?
Mobile phones are portable, but alas, we must plug them into the wall to charge them at the end of the day. 
Noobian:
Converter good
Battery good
Charger good - Just confirm with the seller whether the battery leads will fit into this charger.
Will do. I wish there was some way I could charge the battery and run the project with a charger, like the original set up with the 5v 10A PSU, but it looks like the LiPo packs need special chargers like the one I linked to. Can the battery be on and powering the project while it's plugged in and recharging, or does it have to be turned off? Seeing tonnes of LiPo fire related content online. Not sure if this is the safest choice of batteries. I basically want something I can leave in my project while it's charging. One of the main reasons I'm switching from NiMh cells is because I don't want to have to take them out to recharge every time.
Devices like laptops and phones, you leave the battery in while they charge, and the the device can be on while the battery charges. That's what I want. The characteristics of a laptop or mobile phone battery/recharge.
AncientOracular:
Mobile phones are portable, but alas, we must plug them into the wall to charge them at the end of the day. 
Good point. You must have DC jack connectors for your project so that you can unplug them easily.
I wish there was some way I could charge the battery and run the project with a charger, like the original set up with the 5v 10A PSU, but it looks like the LiPo packs need special chargers like the one I linked to.
It would have been possible to do that with your original setup if your current draw was less than 2A.
RC LiPo needs special chargers because they need to be balanced.
Can the battery be on and powering the project while it's plugged in and recharging, or does it have to be turned off?
Not Recommended.
I basically want something I can leave in my project while it's charging.
This Li-ion pack you can leave in your project. No problem, you just have to plug the charger in when battery gets empty.
Devices like laptops and phones, you leave the battery in while they charge, and the the device can be on while the battery charges. That's what I want. The characteristics of a laptop or mobile phone battery/recharge.
Actually they're not taking power from the batteries while it's charging, instead they simply share power from the input, which is why they take longer to charge while you're using it. The first charger you linked had this capability, but it couldn't give more than 2A.
Let me just check for some other options and I will let you know soon. 
Noobian:
This Li-ion pack you can leave in your project. No problem, you just have to plug the charger in when battery gets empty.
Yes but the whole problem with the LiIon pack was that the boost would be too inefficient, which is why I am now looking at the LiPos to step them down to 5v. Did you mean I can leave the LiPo pack in the project while charging? I keep reading that you should never leave them charging unattended and that they should be in some kind of ammo box or other fireproof container. I could fit the protective container into my project but it seems like most people take the LiPo packs out of their RC devices to charge them on cement outside for risk of fires and explosions lol. Doesn't sound ideal.
I appreciate all the help. This definitely turned into an XY problem.
I think one of these might help. You can use ordinary 18650s with this device it seems. Need to look into it.
What if I build my own packs of 18650s? I could use 3 cells in each pack wired in parallel, then put two of those packs in series giving me a total of 7.4v and 9A, charge it with a 12v 10A PSU then just step it all down to 5v. Not sure how I would go about balancing all the cells in that case.
Take a look at the new schematic I attached and let me know what you think. Would I be able to have project on while batteries recharge / power the system directly with PSU, in this case?
You can use the balancer I linked to earlier, but it would only take in 1 set of series.
Yes the schematic is fine (except for the balanicng circuit + over discharge protection board part ) and you will be able to power the project while charging.
You can rip out a balancing circuit from an old laptop battery pack, or you can use the battery pack as such.
Has anybody tried the Lithium Charge Protector Boards on e-bay with the s-8252 chips ?