You can take your bike on Eurostar to Paris then transfer to a TGV to go further south. Not as convenient obviously.
Have you done that?, It was a real pain when I did it. Maybe it's changed since.
There was no charge for bike on TGV but I had to take wheels off and put it all in a bike bag and put it flat in large luggage rack in centre of carriage, not by door. That maybe a problem if bike is at all fragile / not protected if other luggage goes on top.
If train is already busy / you are not at the starting station like in Paris the rack could well already be full like when I put it on train at Avignon (train had started further south). I got on train and left bike in doorway which was already a storage area for bags... No choice as I wasn't going home otherwise. Guard threatened to fine me 100euro if I didn't sit on floor/stairs next to bike as it was a hazard.
I think another Brit had a compact road bike and managed to put it in regular luggage area on direct Eurostar train to Avignon... I would try that option. Others on this forum may know how doable that is these days.
There are two Avignon stations, direct Eurostar station is on edge of town. TGV station was a bus ride out of town, buses are okay about taking large baggage. (Assuming nothings changed since I did it 6 years ago). Studying baggage sizes / policy is a good idea, sometime you may get conflicting info on different sites. Good luck