Buying a Carvin rig is a slightly better suggestion than, say, taking a bushel basket full of dollar bills and throwing it off a cliff. If you are doing an acoustic duo in a coffee shop, they would get you by.
There are better ways to spend $4K.
Renting is not a bad suggestion. Is there a way to budget $500.00 to rent a PA for the gigs? This gets you a better system, someone to run it, and does away with issues like maintenance, storage, and transportation.
It is my experience that everyone who claims that they are willing to help with the laborious task of moving a PA are always busy at crunch time.
ANY system you buy/rent/borrow will need someone with some sort of clue as to how to set it up and run it.
For something inexpensive to buy, look for something like a used Community CSX60/CSX70 setup driven with QSC RMX or PLX, Crown CE or the like. You can get 4 boxes and 2 amps within your $4K budget. Figure $500 each (used) for the boxes, and $1000(new)for each amp.
The CSX boxes are not my favorite, but they are loud as hell, are incredibly efficient, so you don't need huge amps to drive them, and they hold up well. No, they won't fill a room that can hold 1000 people with concert level sound, but they will rock a fairly good sized dance floor.
A powered box rig like the FBT MaxX9/MaxX4 would cost about the same. It would sound better, but not be as loud.
For reference, an EAW LA rig with power runs about $4K per stack.
For a mixer, a Spirit LX-7 is under $1K for 24 channels.
The DBX 1231 EQ is generally recognized as the cheapest EQ that's worth a shit. Get them for about $350.00 each.
It's up to you to decide how you use the advice in this thread. I guarantee it's worth every penny it cost you.