You can buy a cheap tent I have 3 viviosun and they all work just fine. I do happen to like my more expensive tent more it is 4x8 silver back tent from htg supply(a lot like gorilla tent). But the cheap tents work just fine.
But that viviosun 4 inch fan and filter are straight junk. I can't even use the filter with the fan in a 2x4 with t5. The fan just isn't strong enough and I think it has gotten worse since I got it.
Cheap tent is okay and I guess since u said u had a 4 plant limit your legal but I still recommend not skimping on a carbon filter unless u have no neighbors or visitors. Otherwise someone might feel the need to get baked and have been smelling your house and decide to come in and take your work. CAN brand filters are what I always use, I've tried others and some work okay but none as well as CAN that I've tried yet.
I know you said 4x4 but everyone always wants to upgrade so if you got room get a 5x5, get a 1000 watt dimmable ballast, an air cooled hood (blockbuster is one of my faves) and a hortilux bulb. I'd go with 6 inch fan (hydrofarm fans work fine) and a can 66 filter.
I know led lights are all the rage now, but your on a budget I wouldn't go down that rabbit hole and spend a ton of money on new tech. I've used all the new tech cmh, led, de cmh and imo hps is still one of the best.
Oh and the reason I said 5x5 is cause of sq ft. What doesn't seem to be much difference only a foot bigger each way makes big difference on sq ft and it's just easier to get better yields with more sq ft
3x3= 9 sq ft
4x4=16 sq ft
5x5= 25 sq ft
That's a big difference when it comes time to weigh your plants as you can only get so much per sq ft.
Also don't be scared of a 1000 watt light it won't be that much on electric bill, but they penetrate better which means better yield unless you get really good at canopy management which is what it takes to yield with smaller lights.