I'm afraid I can't help on the bike front rich, but I went to Avignon from ST Pancras last year and it was superb. I find Eurostar very helpful and especially if the train has some space I imagine they would be far more flexible about letting you take your bike than any plane or TGV. After all the trains are just much bigger and better staffed.
The convenience (especially in Avignon), luxury (it's like being on the Orient Express- they ply you with food and drink the whole way) and overall ambience made it one of my favourite holidays ever.
Once you get to Avignon there's a hotel literally on the station which I found really good. I'm sure they'd provide secure bike storage - they were very helpful generally.
I travelled to the local sites using the local trains which made it even more fun but cycling would be even better. Wherever you are you can see the mighty Ventoux looming up.
Orange and Arles were my favourite towns,although Avignon itself and Nimes are very interesting. Beautiful, historical, laid back, fantastic food and wine. I met a big party of US cyclists touring there, who were having a great time.
Not sure that answers you'r questions at all! I would say that if you can get your bike on and can afford London to Avignon it is a unique experience from entering St Pancras station onwards and for me was one of those "once in a lifetime" things - it was that good.
Incidentally I went very late August (the last week the service ran, and it was virtually empty and I think a bit cheaper)
