You're assuming everyone is subject to US laws.
So you think the US is the only country in the World that you can sue someone for negligent design of a product ?
Completely daft.
Moot point. Noone is getting sued in most countries over a custom made activity tracker.
Why donât you spell it out then, through a formal specification.
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I don't know what you mean by formal specification. I need a device that monitors chest expansion and alerts me when it gets below an adjustable threshold.
Define what these are and how you expect to measure them. What accuracy is needed, what ambient temperature, environment, etc is it going to be exposed to. Then you need to tell us what the electrical specifications/limits are. Then there is product life, is it a hour month year, etc. Consider what happens when the dog jumps into the water. There have been many responses to your vague questions like it is a secret project. Is it a secret project?
There are chest strap breathing sensors out there that feed raw data into a recording device, like Go Direct respiration belt. I don't know it's accuracy but it's specifically made for respiration monitoring so I assume it's good enough. The environment will be 98°, moist but not wet for the belt, and room temperature dry for the rest. Electrical specifications will depend on the battery used by the builder, 16 hours of operation will do just fine, although less is acceptable if I can just replace batteries. It's not a secret project.
Did that kind of things for humans and horses. For humans it's nice to have a ECG amplifier - breating is a sideproduct that all ecg amplifiers give you. For horses it had to be a strain gauge, but that worked quite fine. Dogs work like horses (used them as test subjects 
). So, doable, but what's the budget?
For a moment, let us not consider the compliances & standards which is very important in case of commercial medical devices.
But even if you are making this for your custom use, the design process almost remains the same.
We can try to use as many off the shelf IC solutions or modules to get it working, but you cannot carry around a Arduino Nano board with jumper wires.
So a PCB, enclosure and related stuff will be inevitable.
The point is, you cannot compare the cost of an off the shelf solution with cost of R&D required to custom make such a solution.
If that distinction is clear, what is your budget?
This thread is going nowhere so I'll chime in 
The problem goes well past that. If you market something in the US that meets the definition of a Medical Device, but is not approved by the FDA for release, in addition to any civil suits you have to deal with from your customer, you now have a problem with the federal government, who takes a pretty dim view of such things.
There are ways for the design engineer to get around this. If it is made clear that they are not the manufacturer of record, i.e., there's a clear paper trail that shows they are only being contracted to design something to a certain specification, but are not the actual seller, then responsibility lies with @ntntnt58 and that's who the FDA would come after.
Most countries either have reciprocity with the US FDA -- they follow the FDA regulations -- or they have similar regulations themselves.
I've been in the Medical Device industry for about 20 years and it's not unusual for the Regulatory Bodies of random countries to announce, with very short notice, that they will be performing an audit of your facility, manufacturing processes, and design processes. You can't run away from the problem.
Being positive
Some quick research indicates that Heart Rate Variability (HRV) correlates very closely with breathing rate when at rest. I know from working in the fitness equipment field that many better quality heart rate monitors can measure HRV, even if their apps don't display it. RR interval, which is used to calculate HRV, is a part of the BLE Heart Rate GATT specification.
@ntntnt58 find an HRM that can measure RR interval, use that to calculate HRV in an app, and you're done. Nothing to build: just buy an off the shelf strap (I think most Polar straps provide RR interval) and write an app.
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Don't get tricked by the RR intervalls from sport devices. These are all fake, as is the pulserate they give you. I had a nice study done ~ 10 years ago by comparing real (holter) ecg to chest belt data (garmin, polar, zypher and some minor brands). It turned out that the RR interval were mediated over ~ 3 - 5 beats and all had a "security feature" built in: from 60bpm upwards the pulse rate given was systematically 10% higher than the actual pulserate. But you only find this "slight" inaccuracy when you attach the ecg leads and the chest belt at the same time - and also feed 2 leads of the chest belt into the ecg. The RR data was random at best. In the end we built our own monitor that's still on the market for HRV supported training
ECG based breathing monitors only work when the subject is relatively still. I have a couple of them already and I can only use them while I'm sitting still or laying down.
Budget is around $1000. I just need the bare minimum device that works, I will make the enclosure and everything else.
No need for enclosure, I can make it myself. Just need the bare minimum. Budget is $1000.
What's the name of your monitor?
I still have the second series monitor on the "shelf of things long sold and gone"
So much for that idea 
Trying to get respiration rates and depths from HRV is getting pretty far away from the simple situation of using a Vernier Go Direct Respiration Belt.
The Vernier Graphical Analysis App can certainly give the rates and depth.
https://www.vernier.com/video/measure-respiration-rate-using-go-directrespiration-belt/
It would appear that the issue is with developing the alarms at the high and low.
You may be able to spend your $1000 directly with Vernier to modify their app as you require.
In theory yes, in practice no. I had a strain gauge built into the belt to measure the chest extension and from that the breathing rate. Problem is that you have 2 kinds of breating: chest and phrenic. the first you can measure with a chest belt, the second not. So you get "funny" curves that may or may not give a breathing rate - you need at least 1 channel ECG to as a hint to see what's going on (the electric vector of the heart moves with breating, but wit 1 channel ecg you have a project ion that may or may not give you frequency doubeling ...). ... well, that's the tip of the iceberg ... please don't forget that you cannot verify the functionalitity of these gadgets without sophisticated test setup. And vendors lie. Always.
Maybe there is a simpler solution.
Typically during a period of anxiety & stress, not only does the breath rate increase but the heart rate does as well.
So it is possible that you could just monitor heart rate which is much simpler to monitor. And there are many devices that offer alarms for high/low and ranges.
And from a larger perspective, aren't you keenly aware when this is happening? So why the need for monitoring & detection?
Once you feel it or feel it coming on, you can take appropriate action.
--- bill
By the time my heart rate increases I'm already gasping for air. I don't feel it when I forget to breathe. I also breathe in a very shallow manner for a long time without my heart rate increasing and without being aware of it.