*Most* of those modules have fake LM2596 chips - many of the fakes are easy to spot in a simple test jig - just check the switching frequency (at pin 2 of the IC) under load. Genuine LM2596 chips run at 150KHz. Most of the fakes run at approx 52KHz. *IF* you can get a fake to work in the inverting configuration, you'll be lucky to get over 300mA out of the negative rail, and they are only good for about 500mA continuous in the normal positive configuration.
A (genuine) LM2596 in the inverting configuration needs up to 5A for up to 2ms to startup correctly. If its supply is current limited at under 5A, or worse has foldback limiting, it is unlikely to start unless you mod the module by lifting pin 5 and adding the RC network for delayed startup from datasheet fig. 23, (also see fig. 27).
You must also consider if the positive rail coming up first will be problematic for your load. You may need to mod the positive module to also delay its startup, or worst case, for particularly fussy loads, add a supply supervisor IC and MOSFET switching in both outputs to only connect the load when both rails are up and stable.