There are two aspects to consider in washing a gore-tex or similar:
1) Whether the detergent will damage/strip the face proofing on the outer fabric (the surface treatment that makes the water bead up and run off the fabric). This is important because when the face proofing stops working the outer fabric "wets out" and then the gore-tex membrane inside it cannot breathe, because it is effectively trying to breathe through a film of water.
2) Whether the action of your washing process will damage the garment, like whether the protracted action of your washer will abrade the fabric.
Point 1 is the reason why gore-tex should only be washed with a pure soap like Tecwash, as household detergents are generally too strong and will damage or strip the face proofing. The face proofing can be revitalised by applying something like TX10 wash-in, but in my experience it's never as good as the original face-proofing.
Point 2 is more difficult. Machine washing gore-tex is the best way to get it clean (and clean gore-tex works much better than dirty gore-tex, cos it can breathe better) but overwashing will eventually wear the fabric at the usual points (seams, cuffs, etc). Just like washing a shirt too many times will wear out the tips of the collar etc.
For a frequent freshen-up like you describe, where the jacket is smelly but not dirty, I'd be tempted to handwash it with just a little pure soap. This is what I do with my hunting kit when I want to avoid the colour fading and the fragrances and whiteners in household detergents. Remember to rinse it thoroughly or you'll start foaming during the next downpour!
Or, thinking about it, maybe your washer has a freshen up cycle that you could use, like a really short wash cycle. Check the washer's manual.
Hope this helps
Rat