My temperature control is like your wife's. So I am not sure what I say will help much. Sweating is rarely a problem. So I do not have much to advise about sweating just after stopping.

Perhaps, you go to fast! Maybe do not get into a sweating pace in the first place. I am very much a "turtle"-- go slow and steady and stop very little. I think this approach works better in cold weather.

Are you using your head (literally). Controlling the warmth of my head and neck do a lot for my temperature control. The first thing I do when I stop is to put on a hat.

And have you tried various base layers? I cannot hike in many nylon shirts or fleece. They make me sweat. I find a very thin pure wool layer is the best. I would think many bike jerseys may be too wind-proof- maybe they do not breath well. I do not like bike jerseys. It may be lycra in them.