So some advice :
You can't book that train on interrail.eu for a few months for an unknown reason. There is indeed a quota for passholders on that train (all TGVs from Belgium). Call SNCF or go to a SNCB ticket counter to ask.
Other options : mainly going via Germany (a lot cheaper), unfortunately due to engineering works around Cologne the journey takes an hour longer than usual -> too long frankly (departure 06:23, arrival in Lucca at 23:29).
You could also buy a regular ticket between Brussels and Lille-Europe for 19€ and then a seat reservation for Lille-Europe - Strasbourg (10€). I'd do that if the passholder quota has sold out. Afterwards it's easy :
- IC21 Basel SBB - Lugano 13:03 - 15:58
- RE Lugano - Milano Centrale 16:02 - 17:17
- IC Milano Centrale - Viareggio 18:05 - 21:58 3€ through ÖBB
- R Viareggio - Lucca 22:10 - 22:27
Of course it'll be a long day :)