If I bought my missus a corolla I wouldn't even be alive to buy the bike!I presume 4 biggies is £4000?
You can get a good car for less than that. Toyota. Corolla. Petrol. 1600cc. Post 2004 (earlier ones burn oil).
Do a search on Autotrader.
Edit to add... I can confirm that it meets @Banjo 's criteria, with a 58cm Specialized or Ridgeback frame - and without the need for any wheel removal!
If I bought my missus a corolla I wouldn't even be alive to buy the bike!
so long as your bike fits in the back with just the front wheel removed