Ignoring what you end up with, I think you are all set to shoot yourself in the foot and wonder how the bullet got there. Answer: You fired it.
The used market has absolutely no guarantees and projectors have a far shorter shelf life than many other electronics due to the extremely hot light bulb sitting in the center of it. That means it's very hot, and has fans running around it all the time. So, it's cooled, then hot, then cooled, then hot. The fans run as long as the projector is on, and there is absolutely no way, at all, to know the condition of a used projector.
Brighter models are hotter and have even more potential for issues.
A projector with no lamp? So, you don't even know if it works when you buy it?
Do you have $300 you don't care about?
In my experience, if you have a very low budget, and care about your money, then you should be looking at refurbished projectors, from the manufacturer. Epson, is very good with their models providing, I believe, a one-year warranty, and a brand new lamp in their projectors.
So, that HC730 is available for $375 and is guaranteed.
https://epson.com/Clearance-Center/Home-Entertainment/c/cc301
The Sony is a commercial projector, not really designed for home video. It has horrendous contrast ratio, meaning that you will never get good blacks from it, and after dark use will be painfully bright, and not very good looking. It has a much better lens than the Epson, with horizontal and vertical lens shift, instead of keystone correction, so it has much better placement flexibility. A cheap $50 lamp may or may not work well in the projector. Cheap lamps are often just that: Cheap. I hear a couple of times a year from someone who got a cheap lamp and it actually melts the projector because of how hot it runs. So, that's a really big consideration when buying a high end projector and putting cheap parts in it. Don't buy a cheap Porsche and put cheap oil in it, you may end up with no Porsche at all. Same with better electronics that need regular service, you have to pay for those better parts.
On a budget, the Sony makes no sense at all to me. You need reliability over performance and the Sony doesn't offer the right performance for your space.
Oh, and manufacturer lumen ratings are typically a joke.