Unfortunately, it’s not just the radiation that triggers a trade evolution but also the act of modifying a pokémon’s “data” (in the process of changing its trainer information). The only other option is to wait a bit and perhaps your haunter will evolve naturally if it’s old enough to do so, but given the strength of his interest and your friend’s willingness to help (not to mention their partner’s willingness to evolve), it’s perhaps better to go through with the trade.
One way to make it easier on your haunter is by being upfront with him about it. Show him the trading machine and explain how it will work for him. Show him a few videos. Introduce him to your friend and their graveler and have him spend time with them. Haunter are fairly intelligent, so even if this sounds complicated, yours will understand more than you might think.
Also, tell him your exact plan. Emphasize how long the process will be, and make it clear that you’re not giving him up. When the time comes, try to keep him in his poké ball until he’s back in your possession to minimize his feeling of being abandoned. The evolution will take place once he’s released, even if you change his trainer’s data back to you. Afterwards, be sure to take him out for a treat, just in case you do have to apologize.
Best of luck!