Voltage dropping for Micro on battery!

So the setup I have right now involves an Arduino micro hooked up to an HX711 load cell amp and an nRF24L01. the load cell amp is connected to a strain gauge/Wheatstone
bridge setup in which the sensor data is wirelessly transmitted to a receiver. Currently I'm using two CR2032 coin cell batteries (3V, 210mAh) to power the system and the voltage keeps dropping. The system only require 30-40mA to run. The batteries when tested (unplugged) produce 5.8V and when plugged in the multimeter reads at 4.3 and the batteries steadily drop to 3.7V-3.5V where the Arduino stops working. I've checked wiring for any electrical shorts, any other possible ideas for what could be causing this drainage?

I think that you are underestimating the current. The RF24 pulls quite a bit more while transmitting.

What is the resistance of the bridge?

honestly, I cant measure the resistance of the bridge due to the current setup (wires are blocked and very hard to get to). I've measured the current draw on multiple occasions and its always lined up to be in the range I stated. I've heard that CR2032 cant supply a steady 30mA so that may be the problem but I'm not sure. I'm gonna be switching out the batteries to Li-Po to test but a coin cell is a better form factor for the application.

That is way too much for a coin cell. CR2032 cells are designed for 15 mA peak, with the system being in deep sleep most of the time, drawing one uA or less.

okay, thanks for the clarification. Would a standard 3.7V lipo work then? the plan is to connect 2 of them in series to get 7.4V to meet the 6V minimum on the micro.

I would use a 4xAAA battery pack. LiPo cells require specialized charging circuitry (a balancing charger for two cells) and shutdown protection against deep discharge. Overcharging and overdischarging destroys them more or less immediately.

The Pro Micro is a poor choice for your system, as it is power hungry and does not do well with sleep configurations. The Pro Mini is much easier to work with and has advanced deep sleep modes.

I'm using a regular micro. The setup is required to be as small as possible, hence the choice for lipo batteries as they'll be a lot smaller than the AAA batteries if i stack them, but thanks for the idea. Ill test out both and see what suits the needs the best. As far as the board goes, any suggestions for the smallest possible board for this case that is efficient?

The micro/ Pro Micro runs on 5 V or less. Whatever are you talking about? A single 3.7V LiPo would be perfectly appropriate.