A move is a big thing, for any of us. Especially because we move for a whole host of reasons - jobs, family circumstances, more room/less room, more outside space/less outside space.
It can be exciting or nerve-wracking, and of course we know why we're moving and we fully participate in the process. Cats don't have that luxury, and so it can be far more traumatic for them.
Given that Dougal is probably still missing Angus anyway, and doesn't have his good buddy to help bolster his confidence, and do that whole "exploring together" thing, will make a huge difference. Everything to him will be in the wrong place. The new house won't smell like home, and it's all a bit strange.
Have you popped the Pet remedy/Feliway plug ins back in? If not, I'd do that straight away, and maybe buy another couple to pop in as many rooms as you can.
I suspect Dougal is spending more time hiding out in your bedroom because that will have the most intense scents of comfort for him. It might be an idea, if you and Stuart are comfortable with doing so, to make that his "hang-out" spot, rather than the spare room. I know it's not a great prospect, but it might even help putting him a litter tray in there too, but leaving the door open so he can explore if he wants, and then putting another litter tray in the area you'd prefer to have it. It's great that Stuart is feeding him, and in the bedroom, as it will help to maintain the trust that was starting to build up between them. Could you try some Zylkene on his food again?
Is he interacting with your dogs at all? In time, and if he can take things slowly, he might feel a bit more brave about venturing out, especially if you can get everything you want to have to hand unpacked as soon as you can (no mean feat, I know, especially when you're already tired and a bit dispirited).
Has he got his cat tree out? If there's somewhere he favours more than others - a particular window or vantage spot for example, you could try putting the tree out there, with some cat treats on it to tempt him to try it out. Give him a few days to get interested. If he doesn't respond, then try moving the tree to somewhere which might be more tempting. Some cats love looking out of the window, whereas others can find it a bit too stimulating, so you might have to experiment a little to find a place that works the best.
If you do have to begin to consider rehoming again, some rescues may put your cats on a rehoming list, while the cat remains with you, and then encourage people to view the cats in their existing homes. I know there are two rescues near us who work this way, but it can take a while to find a suitable place, so understand your fears.
I don't think you're being cruel. You'd never have asked for advice on this subject unless you'd given it your utmost thought. Most people who no longer want their cats simply stick them on Gumtree or similar and flog them for a few quid, or do the "free to quick home" style of thing, never knowing where their cat/s end up. You're not doing that. You desperately want to make things work.
How old is Dougal? I've had a quick glance back through the posts but I can't find it, apologies. The older he is, the more tricky he might find it to adjust, unless he's a real golden oldie - they often just want somewhere warm and comfy to snooze. I think from the sounds of it, Dougal isn't at that stage yet.
The other possibility, which is not quite as mad as it will undoubtedly sound, might be to get him a cat companion (in time). This would be a case of trying to match the potential companion to Dougal's personality, so that you got a good match. It may help to bring him out of himself, but it's also a tad risky, because you're a busy individual, and introductions often take time to supervise and effect.
I'd say it's early days yet, but the fact Dougal is at least coming out to explore when you're around is a good thing. You could also try spraying Feliway Spray or Pet Remedy spray around the house, on the sofa and soft furnishings, or alternatively, spray it on a towel or soft blanket, and pop that somewhere inviting for him.
Let us know how you all get on. Other than that, is the house living up to your expectations, or is it all a bit over/underwhelming? It can take time for everyone to adjust to a change of surroundings, so take it a day at a time.