But getting back on point, if the house techs you're dealing with are too lazy to write out the mix then you'll have to write it out yourselves. If they don't give you enough time to do that, then take a picture and transcribe it to paper. With the paper in front of them, they'll likely reset the mix properly.
If that doesn't work, gotta hire your own tech. Generally a good idea anyway with those monitor needs.
P.S. If the multi-instrumentalist is the main problem, you could invest in a 4-channel splitter and a small mixer, which would allow the player to balance the monitor mix of his instruments to minimize the hair-pulling. Each instrument would have its own line to FOH, and the HST mixes them in the mains to their hearts content and does not put those channels in the monitors. Instead the player mixes them on stage and sends a 5th line that is for monitors only. The HST would still have the channel EQ to grab any hairy frequencies if necessary. Doesn't eliminate potential problems but definitely lightens the load for the HST.
If you invested in a decent mixer with swept mid EQ etc then maybe you could just send the one line to FOH for mains and monitors, but some HST's will be apprehensive about it. A used Yamaha MG16/4 would actually cost less than a lesser mixer + splitter + cables. I wouldn't mention it except you've got HST experience so you can help dial them in communicate effectively.