If you want to recreate what might have been considered a bobber, and do it for the least amount of money, it is best to find a decent running bike from the get-go. If you can find one where you know the sludge trap has been cleaned, and ideally, replaced, this in and of itself will save you quite a few bucks. Rebuilding the triumph motor requires quite a few special tools. Not too mention, a few whitworth tools for the earlier '60's models. If you want it to last, and don't know the history of the motor, you really need to have it stripped down and the sludge trap cleaned.
In regard to the hardtail...don't weld to the stock frame. Not only because you (or someone else) may want to put the stock rearend on, but also because quite a few builders feel that welding to cast material isn't the safest. There are quite a few really nice bolt on rear-ends out there now for around $250.
Find a nice unit construction, pre-oif bike and throw on the hardtail and a solo seat, and there ya go.
Cheers
JD