Hi
Good for you getting a rescue, you don't say how long you have had him and he may still be adjusting to a new routine. He probably has been able to poo in his kennel first thing, when the dogs start waking and the rounds start early, Unfortunately in kennels you can't go in and let all the dogs out as soon as you are in, so there are somethat have to wait and will have been jumping around for a short time and need to toilet. It can also take several months for everything to get into place, there are a lot of adjustments his system has to go through, mentally and physically.
I am presuming this is what you mean by 'doesn't make it through the night' and not that he is waking up hungry!!!????
You are not feeding him too late, but I would suggest you feed him before your walk. Half an hour beforehand, which will let his food settle. Walking after feeding, gets the system moving and will encourage things through from the last meal. Sniffs and smells also promote dogs to poo!. You may find this will be enough to get him to go either on the walk, or when you take him out before bedtime for a short wander. Introduce a command when he is doing the 'deed' and then you can get his toileting on que.
What is probably happening is you are coming back from your walk, feeding him and he is lying around with a full tummy. He goes to bed the system starts working and sometime early in the morning as he starts to stir, he needs to go!
I would also do two meals roughly the same. Is he underweight? As if a dog is underweight it is good practice to feed them at nightime, so they are not 'burning off' food during the day. It is notriously hard to get some dogs to keep their weight in kennels and this could be why they have told you to feed the larger meal later. This may not be the case, but if so, you will however find that in a home he will not be quite so busy as in kennels or stressing out. Staffis are breed that don't do particularly well in a kenneled environment and he will put on weight anyway. If this isn't the reason they suggested feeding the larger meal, then feed two the same as suggested.
Whatever you do don't tell your dog off for toileting inside, be patient, it will happen and adjusting food times will be a way of trying things out.
Although you may be tempted, I would avoid changing food at this stage, your dog has enough to deal with at the moment and he is best staying on what he was having in kennels for a while, as his system will have adapted and adjusted to it, so hopefully be utlisiting it the best way it can.
There is a lovely book all about coping with a rescue and some of the problems and joys you will encounter, by Gwen Bailey
The Rescue Dog by Gwen Bailey
Gwen was one of the first behaviourists to concentrate on rescues and rehabiliting them whilst in the rescue and in their new homes, she has a wealth of knowledge. Now all the other rescues have followed her lead and they all have people who can give advice, so don't forget to ask Battersea too if you are concerned, your question is perfectly sensible and their behaviourist will be more than willing to help.
Kate