I don't like a "help me plz" when I have to hunt down the pin assignments myself. So I didn't bother. They're unreadable in that fuzzy Fritzing.
But, if you are feeding 6.4V into the 5V pin of the Arduino, you are exceeding the maximum allowed voltage of 6.0V. If it's the VIN pin, 6.4V is barely enough, or not enough especially as the battery gets low.
The HC-SR05's are 5V devices too. So yes, you have some power problems. Also imagine how the battery voltage might sag if one or both relays are on at the same time.
If you have power for the strips, why are you running the system on batteries anyway?
You say the strips are powered from a USB, but you also say "I think the obvious conclusion here is that the LED strips are drawing too much power when they are turned on"... well that makes no sense unless the USB supply is too weak. Then the obvious test is to connect them to a better supply!
Where is this USB supply in the diagram? I don't see it anywhere.