1) can I leave a bag at the start / finish
No, we're starting at McDonalds, just before it opens BUT I'm driving there and parking in the station car park at 05.15ish, so can meet you there and you can leave your bag in my van... we have a cunning plan to enable you to pick up the bag if you finish well before my best case 21:30 eta at the arriveé. I'll message you my number to confirm.
2) what are people's experience with southwestern trains and bikes?
Generally very good, the guards are pretty chill, and the train is usually half empty by the time it gets to Winchester (although it is the Friday before the bank holidays might be busy to Bournemouth), if there's more bikes than the 3 bikes the space is designed for, you can just stand with it in the corridor there: there's actually room for about 5 or 6 in extremis. and there's 2 bike coaches. Bike carriages are in the middle of the train.
what time are you leaving London? outside rush hour you'll be fine - 4-6.30, the above applies. Sometimes train apps on the phone have the platform before the board so you can get a head start on the crowds.
if you were on a later train because you couldn't get your bike on the one you were supposed to be on, I'd be amazed if they made anything of it when you explained.
Just check there's no bus replacement service on the line on your return journey, they are not so chilled and never let bikes on the busses ;-(.