The more I think about a crossover, the more that the 8th Doctor stands out. Brigadier General Adrienne Kramer (of the Washington DC branch of UNIT) would be useful - he needs to get into the SGC with appropriate authority and she can set him up. This places the timeframe for the crossover as being soon after “Vampire Science” (by Kate Orman / Jon Blum). The initial scene of the story would feature their interactions, and the setup of his cover to get into the SGC as “Scientific advisor”, using the name “Doctor John Smith” no doubt.
The tone of the story, though it starts with a purely Dotor Who scene, is Stargate all the way. He joins the team, acting as “herald” and “mentor”. In some ways he’s “shapeshifter” as well - they arent sure he’s on their side, and convinced that he’s not told the whole story. From their perspective the story begins a “raid the base for tech” and ends up “lets blow this thing sky-high and get out alive”. Finally the Doctor needs to be revealed, but only at the last final battle for the gate home, when all has gone to a hot place in a handbasket.
Booby-trapped DHD - with optional planet-buster bomb (batteries not included) - having finished dialing the gate the team sees symbols lighting up in order around the surface. Obviously a countdown, and looking inside the DHD, the implications of it hitting zero. The gate will be destroyed, and if open at the time, send a shockwave back through the wormhole to destroy the SGC. With the gate closed they’ll all die. In the purely stargate universe this would be endgame for SG-1. With the crossover, though, the Doctor would shoot Compassion a meaningful look since she could step into the vortex and escape the explosion.
SG-1 would be dropped off at a planet with a gate, returning to the SGC without Doctor Smith. Mission reports would list him missing (presumed dead) in the explosion that destroyed the gate on P3×19-A.