The fact is that either way you are going to need a 1 phase to 3 phase convertor of some kind, my take would be to obtain a 220/240 volt motor or get the existing one rewired for 240v, if you are sure of someone who can do the job reliably.
This would mean obtaining a VFD that would be capable of running a 7.5hp 3 phase motor on 1 phase, many manufacturers limit their single phase operation to 5hp, some such as WEG I believe offer 1 phase for this size.
You show that you are in the US so the mains frequency is 60hz, a normal 4 pole induction motor cannot run exactly at synchronism or 1800rpm on 60hz due to something called slip frequency.
It will normally run at 1740rpm or thereabouts, unless it is not a normal induction motor which would be very unusual.
In any event, it is probably not going to be cheap.
Induction motor rpm is the freq of the supply, 60hz, x 60 seconds for rpm, divided by the number of pole pairs, in your case, 2. minus the slip frequency, which is 4%~7%.
A VFD would allow running exactly at 1800, or even double the input frequency up to 120Hz.