We would recommend a three-row vehicle, as there are very few cars in which the second row of seats folds flat or can be removed. Two cars in which you can remove the third row of seats to liberate a large flat area are the Renault Triber and the Citroen Aircross X.
Both are great options that offer comfortable rides and great space efficiency. The Aircross X is larger, more powerful and from a segment above, but the Triber is more affordable. Also it has a lower boot lip height, which will make entry and exits easier for your dogs.
All things considered, though, the best all-round option for you is the Kia Carens Clavis, though it is more expensive than the aforementioned two. Not only is it a much better-rounded vehicle with more powertrain options (including an EV), it’s also the most practical.
It has a flat folding third row, which on its own frees up lots of space. Moreover, if you need more still, the second row will fold near-flat in the 7-seat version. You should be able to fit a mattress for your dogs with a bit of folding.