Quite frankly, I am shocked that little machine shop sent out a replacement tailstock. They should know better - the tailstock is matched to the lathe.
I would try very hard to get a new lathe. Realigning the machine, even to import standards, is a very involved job. You should not just chop a bit off the bottom, it must also be free of pitch/yaw which is harder to measure and correct. Even if you are happy with the results, it is unfair for you to repair a machine that should have been working from the start.
I would be extremely cautious about touching the headstock. I'd hope the headstock is currently aligned to within 0.005"/ft of the ways and if an amateur starts moving things around the machine will become irreparable garbage very quickly.
A metal lathe is a precision tool and alignments should be orders of magnitude more precise than a person can verify by eye.