An option could be to drop the car at a Local car park. They are a bit cheaper out towards Hounslow and jump on the tube (Hounslow east to Hammersmith is about 15 / 20 minutes)
Parking is free on the streets in and around South Ealing / Northfields. This area is 10 mins on the tube from Hammersmith and it’s very close to the A4 (M4) it’s also safe.
There is a lot of parking in Hammersmith but it’s pricey and with the bridge out it’s a traffic nightmare.
Feel free to PM me for a bit more local knowledge if needed.