Help! Unwell baby only sleeps on my chest but wakes up when I put him down

My 11 months old had a fever last night reading 39.7 at 5am. Since then he did not sleep all morning and was cranky. He didn't have his naps only 20mins nap while he was sleeping on me. Soon as I put him down he wakes up crying. I saw my GP today to check if he has any infections. She said his ears are slightly red bit nothing to be concerned about, he has runny nose and mucus on top of this throat. Chest is fine and to comtiune giving him water. He has been avoiding food today but drinking milk as normal. Since his temperature was one off I am so confused why he is cranky I get it he hasn't sleep all night and day. But he keeps crying and starts hitting me out if frustration when I put him down as he wants to be carried. I haven't slept since 5am last night only had 3hrs sleep and am absolutely shattered. His asleep now on my chest but soon as I put him down he wakes up crying I can hear his breathing changes when I put him down on the pillow there's a lot of mucus in his throat and probably a slight blocked nose. I even tried raising the pillow but he wakes up any sound and little movements. I dont know what to do i thought he might be in pain as he got a cold aswell so i gave him calpol and i put teething gel on him (he has been teething forever) since he keeps bitting.Please tell me how I can get some sleep i can't sleep with him on my chest his a big boy now and it's uncomfortable. I'm super tired and feel over stimulated having a baby glued on my chest non stop the whole day. Husband helped after work but baby only wants me.