You'd probably be better off waiting the 4 months (or to be more specific - setting the closing date as after the 4 months; using the intervening period to go through the sales process / deciding on your next purchase etc.).

If you sell within the 2 years - you don't qualify for the FULL exclusion upto $250K single / $500k married. You may still be able to claim partial exclusion if you had

a) a work related move

b) a health related move

c) Other unforeseeable factors.

https://www.irs.gov/publications/p523#en_US_2021_publink100073096

See the link above for more details on these criteria for partial exclusion.

While the accounts take time, new pulls show up almost immediately (a day at most); multiple pulls in a short while may result in rejections.