So, my 2cents is, if you are tall, go for the deeper one!
I have not tried a La Bassine. I have tried an installed birthing tub in the birth center, and then a fishy pool for the last birth which was at home. I also have long legs at 6'0", and after my experiences have chosen to get BPIAB.
The birth center tub was nice, but I actually preferred the room of the fishy pool. HOwever, neither of those was deep enough to cover my lower back when I was on hands and knees, and I desparately needed that during my more painful contractions. It still worked out ok because I had my husband pour the warm water over that area during a contraction, and I would certainly go for a fishy pool again rather than the birth center tub or nothing,
BUT in searching for this birth I was definitely looking for the deepest tub I could get. BPIAB seems to be the one. (I suppose Aqua Doula might actually be deeper, but the cost was prohibitive, and renting one would cost the same as buying BPIAB, so I'd rather own my own.) We bought it and set it up for a trial. I think it's going to work out great.
If height is not an issue, then I bet the La Bassine might do just fine -- especially if the fishy pool worked ok for me. The only thing I'd say between the La Bassine and the fishy pool, is that the fishy pool definitely has a tendency to spill over, and i hear that the La Bassine's sides are stiffer, more like the BPIAB.
Another thing I like about the BPIAB is that the height is adjustable because of the three ring design and so my shorter sisters can use it and take the sides down a bit if they need it.