Heat, sun and baby Entertainment

I always used to put a bit of baby suncream on any bits that might end up in the sun when walking somewhere, usually feet. Obviously the best thing is to keep them in the shade but realistically that’s not always feasible. At 10 weeks we went on holiday to DH’s country which is very hot in the summer and she was just fine! Like you I figured better a bit of sunscreen than risking burning. And I wouldn’t worry too much about entertaining them, at that age they don’t need a lot of stimulation.

I think the advice is to try and avoid sun exposure until the dc are over 6 months, so that may be why you have been advised to avoid suncream. However, if you are going out and your dc has a shade and parasol (so not in direct sunlight), you should be fine, although I would continue to put suncream on just in case.

Try having a bath with him during the day. Put him in his baby chair/seat/bouncer whilst you get in, and sit down before bringing him into the bath with you. Have it at the recommended baby bath temperature.
You could also try laying him on a water mat like this for tummy time, as if you fill it with cold water it will cool him down. You need to supervise him at all times obviously (the product says 3 months and up, but with supervision your ds should be fine).

No advice but just want to sympathise. I had a baby who couldn't be put down and had to be walked around constantly to stop her from crying. It's relentless, boring and exhausting! My baby was born this time last year in the UK in the middle of a really hot spell. I remember walking outside at 4am with her in a sling. Thankfully we live in a cool house but my god it felt like it would never end!