Beating the boss for the first time will take a while, but you only need crowns for the really big upgrades. Most upgrades only cost XP, so you should be able to upgrade your tower and troops and income to get to that point.
I tried it again. Firstly, it didn't save my progress from before and I didn't get the upgrades like I did my first playthru.
Second, the upgrades don't even tell you the cost and I don't see a gold tally up there.
Furthermore, why do Knights cost more each purchase, even when there are none on the field?