3.5 ported...
In an 18" format look for a Qts of 0.25 or lower..... good luck with the 300 dollars unless its an entry pro woofer with next to no throw - which may work too.
most of these are about 1 grand. What you're asking for (if done right) is not cheep.
TC Pro 5100 Neo
Sound Stream XXX
Fi BTL (ask for soft spiders)
old school TC-5200 or Eclipse Ti Pro or any 4HP
IA Warden (again, ask soft spiders or the Q will be too high)
custom DC might work too, i think they have some super neo thing too.
B&C SW115 Neo
The second issue is a big enough port... You'll need a huge port and it will have to be long in that size box. Unless its going to stick out of the box a few feet then you probably have to do it with passive radiators.