The concept of "claiming" extra space relates only to the listing you put on the house when you go to sell it. You can "claim" anything you want, however I suppose you could be in violation of some listing regulation if for example you "claimed" that you had a 2500 sf house when you really had a 1500 square foot house.
Speculating on how the particular tax assessor in your town is going to handle an improvement to your property is possibly amusing, but no one on this forum can possibly know how it will really pan out. When I put a deck on my house, and pulled a permit, I inquire as to exactly how this would affect the assessed value of my house. Turns out that in my town, a deck is assessed at a fixed price per square foot, regardless if it is made of gold plated timber or ordinary wood. The assessor does not even bother to look at the finished deck, they simply take the size off the plans filed with the building inspector.
Now if you "forget" to file for a building permit, then the assessor will presumably not find out about the deck, at least until you go to sell the house. Then they may determine you have an unassessed improvement, and will bill you for back taxes, plus penalty. Or maybe they will not notice, and you will escape the tax man.
I have no idea how your town handles things, and neither does anyone else who does not live there. As for your contractor arguing that you can "claim" extra space because you now have an egress window, in my town we are on septic, and the required size of the system is a function of the number of bedrooms in the house. So the issue of whether a basement has a legal bedroom is very important, and the definition of a legal bedroom is pretty explicit, and it includes more issues than simply having a window, there are minimum height requirements, a privacy door, a minimum number of square feet of window, and a few other requirements I cannot remember. But if you improve your house and go from say three bedrooms to four, you typically need to upgrade your septic system, unless the system is already sized for four bedrooms. If you are on sewer, this does not apply, but there is no sewer in my town, so it is an important consideration when planning an upgrade.