Yes, approaches 3 and 4 certainly make the most sense among those I listed. The choice of headphone amp would probably be driven primarily by the type of headphones you choose, as well as by how much you want to spend. I have no particular suggestions to offer, assuming you are interested in conventional dynamic headphones, as in my case I use an older pair of Stax electrostatic headphones. Those require a special purpose amplifier suitable for driving electrostatics, which is also made by Stax in my case. And Stax or other electrostatic headphones tend to be quite expensive these days.
Yes, using the LM just as a power amp (connecting to its "pre-in"), with a separate preamp driving both the LM and the headphone amp (and preferably providing a "tape output" that would be used to drive the headphone amp) is certainly a viable approach. But of course it would be more expensive than approaches 3 and 4 (unless you can find a headphone amp that would provide such a preamp function itself), and it would result in sonics that may or may not be either better or worse than those approaches.
Good luck. Regards,
-- Al