I wouldn't dabble with welding, if it distorts and it will you'd have to send everything to stress relief and machining, that's over your budget. T's and H's construction steel are very bad against twisting forces so avoid using them 'as is' all together.

If you want to build a router you probably wanna build with the biggest and heaviest aluminum extrusions you can get, something like 80x160 or 90x180 or larger. If they have big holes in the cross section even better, you can pour some local non shrink grout with some gravel for damping.

You want ballscrews(size 20-25), linear rails(size 20-25) and closed loop steppers(nema 23 or 24) with drives running 80-110V DC, or cheap but good ac servo's(60st) like lichuan from aliexpress(you can bypass aliexpress and buy from them directly too). For the controller you have a few options available depending what you want to run, mach, uccnc, acorn etc. I'm going with uccnc and their axbb board for my build, particularly their ready-made electronics box.

Do you have weight constrains? Best scenario would be to get a precision granite plate and build with extrusions on top of it. It would be the easiest, lest prone to error and give you the best results.