@JessieLongleg That sounds really tough and your husband sounds like an utter plonker. Good lord, could the man not take 2 minutes to fill the baby bath for you?? What an ass.
How is your son if you play with him on the floor? Does he need to be in your arms all the time, or will he lie on the ground/his cushion and play with a toy while you are there to hold the toy, reassure him etc?
And how is he in the highchair if he can see you in the same room as him with you talking to him and playing with him?
Your end goal is for him to be happy on his own for a few minutes while you get things done, but you may need to work towards that in tiny steps.
If he is happy to sit in a chair / lie on his mat with you in the room, can you pop out of the room for a second and then come back again before he is upset? If that works, you can gradually build up to longer times out of view without him getting upset at all.
If he is only happy in your arms, can you work on being able to put him somewhere comfy while you continue to touch him and play with him? If you pick him up again before he gets upset, he should gradually feel more comfortable in his seat and you can build up to less and less contact.
Also, you can also get a lot done with him in view , so if you get to this point things will already be way easier. My DS (4.5 months) likes his highchair, so I move it around my flat (appreciate this is less easy with stairs!) and put him in when I need to do things like showering or getting dressed. He's much happier in the chair than being put on the floor/bed in the same room because he can see what is going on.