As far as I'm aware, that's usually how it is (though some of the Lay-Z-Boy - or whatever they are - models can be had for under your budget). For what it's worth, I spent a bit of time myself on this subject and ended up loading out with a Wheel Stand Pro and a good-old-fashioned bean bag (although space was a factor for me, so this might be a more appropriate for people reading my response with that concern)!
I can adjust it how I want and it set me back only £65 for the bag, with beans (the stand was £110). The bean bag does require 'puffing' for a comfortable sit; but get it right and it's as good as any racing cockpit out there (wanting perhaps only extra height for those that require it)! Getting a good-sized bean bag is important though, as you'll likely want to be comfortable and have support where you need it (such as the back and neck). Plus, they're also versatile, which is great if your set-up requires a bit of flexibility and/or space-saving.
Otherwise, seats can offer the rigidity and convenience of just sliding into your cockpit, but I too was put off by the fact that you'll have to pay up big-time to get it.