I concur with Mark's (
@mksj ) perspective on size and model. I have the PM-1340GT and found it to be a very good size, even though I was originally trying to special-order a 1324 from PM. I have also read enough here on H-M to understand the value of a 3-phase machine in terms of surface finish and the ability to control it with a VFD and all the benefits that this can provide. The Acra BV-1340 looks like a very nice choice, but it's outside your stated target budget.
One thought I had was back to your original thinking of the Eisen 1340 as a starter machine. It's single phase, but you could always do a 3-phase conversion downstream when you hopefully have more money to invest in it. I replaced the factory PM-1340GT 3-phase motor on my lathe with a vector-rated replacement and a new set of controls from Mark that also provided electronic proximity stop, joy-stick jog function, better E-stop, coolant control, and of course variable speed. The vector-rated motor provided enough additional torque and power that I could dispense with a 2-position belt drive and stick with a single belt position. I don't know what Mark would charge now for a new control package, but his estimate of about $500 if you did it yourself sounds generous. A replacement 3-phase vector-rated motor would probably be in the $600-700 range - it all depends on the physical size limits which dictate which models you can squeeze into the existing mounting platform. They do pop up at New Old Stock on eBay from time to time (I got my $1,600 Baldor for $500 this way, but you have to be willing to wait). Have a look at
this thread. And
this $570 Marathon unit is one of the motors that is typically substituted for factory on this size lathe, but there are others. Anyway, you could start with the single-phase Eisen, and over time switch to 3-phase if you have the motivation and cash.
As for tooling, don't let others bludgeon you into shop-made HSS tooling if that doesn't sound attractive to you. I'm on my 3rd and final lathe now over a 25 year span, and have never ground any HSS tooling for any of them. If you absolutely want the cutting characteristics of HSS, you can always outfit your indexable tooling with HSS inserts. You'll find that discussed on page 42 in my book - which apparently you already have.
AR Warner makes a number of indexable HSS alternatives.
I'm attaching my curated list of other tooling you might discover you want to consider at some point. And at some point you'll be scratching your head over what collet system to settle on for the lathe, so I'm attaching a write-up on that topic as well. If I can help further, feel free to DM me.