Steve which Harrison is it, 11", 140 up to circa 1965 have clutches. 155, 165, 12", 13" and 15" and newer have spindle start with latched in foot brake, both would need wiring/figuring/linking in to a decent inverter, choose wisely.
My Direct Drives inverters 110+ page manual is absolutely useless gives everything apart from tell you what the terminals are, not much better is the aftersales.
Should have gone this route 5 yrs ago **LINK** DD cost more than that for 4KW Inverter and £280 repair bill after 1hr 27mins! But still work out all the interlinks footbrake, spindle forward, spindle reverse, panic but manuals cant be worse than above name.
Looked at various older lathes even the two speed ones reaching 1700rpm, most are 1000rpm to 1400rpm.
Having run a 140 rare 1ph for 9 years before scrapping I wouldn't advise of single phase. Although motor power via twin belts to clutch, the two cylcle pulse of a single phase motor is transmitted, you will never get a decent finish. 3ph motors cycle 4 times per rev as against 2 cycles per rev for 1ph.