So I changed the bottom bracket, crankset and pedals on a bike. I had measured the chainline around 48mm before installing the new crankset. Even though the SR Suntour crankset I installed stated 48mm chainline, I measured it at 49mm (by sticking a piece of paper between the ruler and frame).

Now, the FD can go to the middle and small chainring fine. However, when trying to make the chain go up to the large chainring, it won't catch. So, is this more a barrel adjuster issue or limit screw issue? Would it be worth it to add a bit of slack in the cable?

Limit screw or cable tension (more not less).

What's the chain line on the new crank?
You need both to compare.

It may be simply adding cable tension, backing out the H screw or installing the correct length BB.

I would start by checking that the high limit screw isn't preventing the FD from moving far enough outboard. You can do this by attempting to shift to the big ring and then manually (by hand) seeing if the FD will move any further outboard. If not, then the high limit screw probably needs to be "backed" out some.

If the chain will shift to the large ring by manually moving the FD, then you need to add tension to the cable. Be careful though as if you adjust the cable to much, then you might not have slack for the FD to shift to the small chainring.

Best of luck. Front shifting can be a pain to set up, especially triples. I guess one reason SRAM is pushing 1x setups.

Trying "this or that" on a system that is in an unknown state is a poor plan, I would recommend a systematic approach instead; disconnect the shift wire (remembering how it was attached) and follow this procedure from the beginning as if it were a totally-new installation: https://www.parktool.com/blog/repair...eur-adjustment

I finally went out to the park to use the public bike stand. So the question had been if the chainline is 1mm longer, would I be able to access the large chainring. After playing around with limit screws eventually I decided to undo the bolt holding the front derailleur cable and add or subtract cable tension with the limit screws being either at the bottom height or middle height. I had the screws around middle height when I finally got it to a point where I could shift to all 3 chainrings and also not rub too much either on the inside or outside of the FD cage. Even though I don't think I had it absolutely perfect, I was satisfied since I was ouitside in the heat and could barely concentrate.

Now I'm satisfied I can roll. I don't know when I'll make a video showing how I modifed a department store bike though. I think it would be interesting for others to see.