Thru the wall AC's are for the most part disposable AC's they are noisier and designed for one room application. Similar to window units but typically larger.

Temperature control nearest the unit is always cooler than farther away from the unit. If not sized properly you could have large temperature swings in the room it is located in.

These types of units are designed to fail. The manufacturers make it difficult to do any maintenance to them, many parts are plastic. If you are handy around the house you could install it yourself and when it breaks just buy a new one and install it yourself yet again.

Due to these reasons above, it is unlikely you will find a HVAC service company willing to work on it.

If the doorways are open to the kitchen and dining room, then it makes it more difficult for the AC to keep the den cool. Can you install some type of door to seal off the den? Even a simple bifold door or heavy curtain would help.

It is difficult to estimate how much it will cost to make an hole in the exterior wall. You want to ability to seal the opening in the winter and make it relatively easy to pull the unit out if it needs to be serviced. Spend the money on a good unit.

This is 80% carpentry,15% electric and 5% hvac,guess who you shoul;d ask about installation.