My baby is 5 weeks old and in a cotton swaddle wrap for sleep- how do I dress him for summer nights without overheating? The wrap settles him but I’m worried it might be too warm even with minimal clothes and a fan on in the room.

Red Nose recommends using a light muslin wrap for swaddling.

Red Nose recommends dressing the baby according to the temperature in the room rather than controlling room temperature.
There is no recommended amount of clothing for sleep for all babies.
Keeping babies head/face uncovered is important as babies control their body temperature via head & face.

Sometimes, on very hot summer nights, baby may only need a singlet & nappy.
In warmer weather, you can dress baby in a sleeveless and legless onesies, or just a nappy and singlet.
For cooler weather, a full suit will keep baby warm.
https://rednose.org.au/article/how-to-dress-baby-for-sleep

Dress baby for sleep using layers as you would dress or use layers yourself: to be comfortable, neither too hot nor too cold - add/remove lightweight blankets to ensure baby’s chest feels comfortably warm to the touch
https://rednose.org.au/article/how-much-bedding-does-baby-need

Consider the following when deciding how much bedding baby needs:
The room temperature where baby is sleeping.
How hot does the baby feel? A good way to check baby’s temperature is to feel baby’s back or tummy (don’t worry if baby’s hands and feet feel cool - this is normal).
Whether the baby has a cold or infection or another special need.
Consider how many layers that you as the baby’s carer are wearing comfortably.
https://rednose.org.au/article/bedding-amount-recommended-for-safe-sleep