Help with decoupling capacitors not working
This is a fairly typical problem for modern radios. If you check the datasheet, all their recommended power supplies contain a 330uF cap. The same datasheet says the module makes impulse loading of 2 A at 5 ms period.

Now consider this: 2 A at 4V is a 2 Ohms load. If you have just a 100uF cap, the RC is 2e-4 s, or 200 us. The load duration is 577 us. This means that your 100 uF capacitor will discharge nearly completely during the RF transmission pulse. Of course the feed from battery will somewhat help, but still... You need to increase the local bypass capacitance at least 10-fold, with low ESR caps, maybe several 100-uF ceramic caps, then you might make it work.