Have you tried clicker training and reward training? Food is a great motivator! I was taught to reward the good and ignore the bad. So give praise and treats when peeing and pooping is done in the right area....and the cold shoulder when he does it wrong. Bulldogs are people pleasers. I'm sure that by sticking to the same strategy...Baxter will eventually catch on that pets, love, praise and rewards are given only when pooping is done correctly. You must give treats every single time it's done or the conditioning will not work.
Then you can end up with a bulldog like mine. He knows that he gets rewarded for peeing outside. Now he has been conditioned to go outside WAY MORE OFTEN than is needed. He's way too smart for his own good.
Nah they get fully trained. I think it's the changes you are going through right now. Once you get settled in again and he can have his comfort zone and routines it will be fine. Raven stopped having accidents around 8 months old. It is very rare that she has an accident and it's not usually her fault. It is mostly on the kids not taking her out when she asks them to. Ty is now 8 months and since his neuter he has only had two poo accidents in the house and that again is on the hubby and my kiddies not doing their thing because they are caught up in their gaming. Raven is turning 4 this year and she has had full run of the house since she has been a year old with no problems. I do feed them on a schedule and make sure that they get their potty time before I leave for work. Good luck.
Always in my heart and never far from my thoughts. See you at the rainbow bridge my sweet little Monkey. R.I.P. Raven.