You're not being unreasonable to consider rehoming home however there are steps you can take to improve things until that can happen.
You would be unreasonable to expect him to go into a kennels based rescue/shelter though. They're not nice places, even the plushest, nicest run options, will be very stressful for a dog.
- Have him sleep in your room. That instantly ups the amount of time he has with company by a good 7+ hours.
- See if there is any local person, teenager etc, who would come and let him out a couple of times during the day for cash. You might even get them to sit in with him and watch tv/use the internet. If you don't ask, stick a card in the shop window, ask on a local FB group, you don't know. Whilst a professional with all the insurance would be better if they don't exist in your area, then they don't.
- Find a rescue that will support you rehoming directly from your home - lots of smaller independent rescues do this, so they will help to advertise him, and find potential homers, they may even do home checks and transport, but he stays with you until that home is found. If the home falls through, they provide the back up so he does not come back to you.
If he is toilet trained when you are actually at home you could look at building an indoor toilet area. To do this, get a puppy pen so you can separate it from teh rest of whatever room it is in, ideally..
Find a mechanics drip tray - amazon have these, they're available in a variety of sizes and are wide, shallow trays that are much bigger than a cat litter tray. Find one that fits the pen size if you're using one.
Line this with either disposable pads or washable ones (washable ones are around teh same price for 4 as a large pack of disposable ones costs!) - then cover that with some fake turf (washable in the shower/bath/hose outside, whatever option you have there).
Then spend some time over a weekend taking him to this area and rewarding him well for toileting there. And then hope that whilst you are out, he chooses to use this area. You may get lucky, if he is only toileting indoors bcause he has no choice, he might well.
Decent rescues are full up, they always are, and have waiting lists. Beware the temptation of selling him in the free ads or giving him away, people do use those to source free/cheap dogs for nasty purposes unfortunately.