Forget the Archangel kit as it will not make the rifle any more accurate, and if you are accurate you don't need to look "cool". Once you have all of the mods done to make it accurate then you can tacticool it out all you want.
Start with the recoil buffer pin. I made one for next to nothing using some vinyl aquarium tubing and a 20 penny nail or you can buy a Delrin one for $5.
The trigger can cost you some $$. I like my Kidd single-stage trigger group with the mag release http://www.coolguyguns.com/Single-Stage_p_29.html
I also have a Volquartsen https://www.volquartsen.com/products/205-tg2000
and a couple of stock triggers that I have worked over myself. The Kidd is the best though.
The scope is important, keep your eyes open for a clearance sale and you can find a decent one for around $100. The more you spend though the better the glass. Many shooters will spend as much or more on a scope as they will on the rifle it is mounted on. It is money well spent. I would stay the Hell away from holographic or red dot sights/scopes. Those are not designed for hunting and they are not as accurate out on long shots as a telescopic scope with a cross-hair reticle of some kind. I would highly recommend a 2X9, 6X12 or something of that nature. Good out to and beyond 100 yards. In th emeantime the BSA will probably suffice for the time being, but once you have some good glass you will never mount the BSA again I bet.
Next I would probably either do the stock or the barrel. Personally I prefer a heavy barrel, again, its gonna cost some money for a good one. I am wanting to get a Volquartsen Carbon Fiber Tension Barrel https://www.volquartsen.com/products/546-thm-tension-barrel-carbon-fiber-sleeve
With the heavy barrel you will have to get a different stock to accommodate it- your Archangel may work with a heavy barrel but you would have to check. Personally I don't care for them though. My opinion is that if you want a "tactical" looking .22 rimfire then you would be better off getting the S&W M&P 15-22.
Just my thoughts on it. Good luck and keep us posted on your progress!