My 3m old wriggles a lot in the night and works his way to the side and top of his bassinet. He is not strong enough yet to lift his head for a long period of time so I am not worried about him rolling yet. But I am concerned about his face being pressed against the bassinet which which has fabric sides not mesh. I originally assumed the sides would be ok when purchased.

Is the bassinet in a well-ventilated room?
While mesh sides are preferable, many bassinets do not come with them.
Remember, that there are no Mandatory Standards for bassinets & often a cot is preferable.

Are you wrapping/swaddling your baby?
Red Nose recommends using a light cotton or muslin wrap for settling baby.

Some babies do not respond well to swaddling & some like their arms free to touch their face or suck on fingers.
Many of these babies will wriggle & squirm in an attempt to get arms free, & often their swaddle comes loose, or they end up in various positions in the bassinet..
You may like to try using a muslin or cotton swaddle but wrapping with arms out as per article below.
https://rednose.org.au/article/wrapping-babies

You could also try him in a sleeping bag (arms out)
If he continues to wriggle squirm, you could also try him in his cot.

Are you using a blanket?
If there is any risk of the blanket covering her head/face then it may be necessary to stop using.

Have you started tummy time yet?
How does he go? - if you haven’t been doing tummy time yet, it is important to start.
Is is probably time to be increasing his tummy in preparation for rolling.
Most babies will be rolling by 4 - 5mths
If he doesn’t tolerate long periods, try to have TT more frequently.
https://rednose.org.au/downloads/RN0407_22_Tummy-Time.pdf

Always continue to place baby on back for sleep.