The video below show what you need, and how to do it.
I have always liked that video. Man, if we all have a steady eye and hand like that guy obviously has.
That is some precision work with what looks to be a Japanese type of chisel.
There are many UTube videos featuring Japanese hand tools such as chisels, saws and wood planes. I get lost watching them sometimes.
The Garden Archways for example. Multiple mortise and tenons done by hand. Simply amazing.
Yeah, you can Jury Rig something up for yourself but always run the risk of messing up decent equipment.
A person has enuff going on all at once with equipment made for the job, let alone always wondering if something is going to fail on you at a critical moment.
Todd is a great option for tip lathes.
At the very least, there is often members selling various lathe parts in theis Forum.
Pieces of Lathe bed, chucks, etc. If you feel confident in putting one together.
Keep in mind that it does require a bit of lathe knowledge to be able to piece one together and tweak it to where you like it.
Unless you are lucky, you can't just toss a bunch of pieces together and it will work 100% first try.
I have also always liked the one that Chuck built. I bet that one works like a Boss.
Better off to spend the money on a ready made one, and then after you see what goes into one, you can play with parts for your next lathe.
You will have countless hrs of fun and get your moneys worth from a ready made without causing yourself any grief.