Couple things bud. I know this is likely your forever home so a future sale might not be in the cards. However, if you ever do need to sell, the purchaser's insurance company will likely want a stove inspection and to see paperwork from a qualified installer. While you can go cheap on a stove there is nothing cheap about a chimney pipe that is installed to code. For example, if you go through a wall you have to put in a double-wall insulated box, if you go outside the house it has to maintain a minimum clearance from the siding depending in the material and pipe used being single or double-walled, extend a specific height above the apex of the house, capped with a spark arrestor etc.
You'll need to find the manufacturer installation instructions for combustable clearance for that specific stove as well as the NFC for standard clearances. Lots of variables, heat shield, wall material, floor material, hearth size, pipe clearance from wall etc.
The material for the chimney alone can cost you over a grand to do it to code.
Here is the one I did at my house. To take advantage of the minimum clearances I built a noncombustible air-gapped heat barrier. I placed cement board on the wall, then metal studs on their side, then cement board, and finished with Italian slate. This posed its own challenges like heat certified thinset and grout. Aka, more cost
The vent at the top and bottom allows the heated air inside the barrier to escape because it creates a cool draft intake at the bottom and hot air escapes from the top. This keeps the interior face cooler. The intake and exhaust size has to be a certain coverage area based upon heated face size. This allowed me to place the stove within like 9 inches of the nearest combustable which would have been the face of the original sheetrock wall.
View attachment 96005