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Sounds like you have very specific time frame? You can probably find more options using slower trains but, they're um slower. We've used the Regionale and/or MD trains to go from Barcelona down to Amposta for an MTB event. You can just wheel your bike on and off them.
Some parts are harder to get to then others, ie. France has trains into the Pyrenees but Spain doesn't really.
What's "the first train from Madrid to Barcelona to connect with the TGV doesn't take bikes."? You mean, doesn't take complete bikes, but what about bagging it?
I don't know what an interrail pass is. We just use renfe.com and maybe the French and DB train sites to book stuff.
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Yes, the train works for me if I can do it in not much over 24 hours. If it takes two full days, that's too much time.
the first train from Madrid to Barcelona to connect with the TGV doesn't take bikes
It's a low cost Avlo train, 2nd class only, maximum luggage size is too small for a bagged bike. The first AVE that you can put a bagged bike on is too late to get you back to London that day.
I've spent hours looking into it for events this year and next.
You can do London-Barcelona in a day but not any further.
Connections to Madrid are good but to the south are not great, eg only one train per day direct Barcelona to Cartegena, one per day from Algeciras to Madrid. Those are not special examples, they're the two journeys I've looked at booking.
Coming back, the first train from Madrid to Barcelona to connect with the TGV doesn't take bikes.
Can't get to France early from beyond Madrid to catch the sleeper trains so need to stay overnight.
Also it's next to impossible to book onto Spanish trains with an interrail pass in advance. Risk is you have to hope you're lucky at the station.
It's much easier to get to / from Italy, Croatia or Poland from London than southern Spain.