I can't see an AAV, but at a guess reckon it's worth putting one in.
Where the cap is on the top/left of the trap, not sure if there's enough room, but one of these:
https://www.bunnings.co.nz/aqualine-40-50mm-air-admittance-valve_p00940727
because of lack of clearance - may need to put a 90 deg elbow to fit it, or loosen off fittings so you can rotate it into place, I don't think it matters if the AAV is vertical or horizontal.
My guess is that suction from flushing the toilet is draining the shallow trap in the shower rather then the basin, but fitting an AAV under the sink should do the trick anyway.