Thank you! As usual, super informative. About the 1x15s: I don't like the idea of one 4 ohm 15, because I might want to add a 2x10 or some other cab, (Which is why I went with a 4 ohm/2 8 ohm 2x12/1x12s for a balance) and I'm wondering about "2" 8 ohms. If the LM3 is as weak at 8 ohms as people claim it to be, I don't know if 1 "15 would be enough. I could probably combine then, like a 1x15 and a 1x12, but I'd rather have two of the same or just one 2x12
EDIT: What I mean to say is that I'd rather have 2 "12 speakers as opposed to 2 "15 speakers. (For mobility as well)
If you're going to be running a single, larger cab (in lieu of two smaller ones) with the LMIII, I'd go with a 4 Ohm cab, no question, whatever its driver complement. I don't know that I'd call the LMIII "weak" at 8 Ohms -- my little CMD 121P combo, powered by an LMII (Combo Head) is astonishingly loud, for what it is, at 8 Ohms -- but it definitely has more to give at 4 Ohms.
In the case of the Hathor 1533, you can switch it between 8 and 4 Ohms. At 4 Ohms, it's as efficient (you get as much sound-per-Watt), as the Hathor 2x12 -- actually a hair more, per Duke LeJeune's specs (98 dB/Watt/meter, vs. 97 dB for the 2x12). So, if you feed both cabs 300 Watts, a single 1533 will be as loud (or a hair louder) than the 2x12. Same deal with 500 Watts. Now, with enough amp, you can go louder with the 2x12 (it will handle 1000 Watts, whereas the 1533 tops out at 500), but since the LMIII is a 500 Watt (at 4 Ohms) amp, with that head, it's not a consideration.
Other things (schlep, form-factor, voicing) might very well still be considerations -- and, with a micro-head like the LMIII, I, personally, would prefer a taller 2x12 over a shorter 1x15. I threw the 1533 out there as an option because of its 4 Ohm capability, because of the LMIII's parameters (500 Watts, 4 Ohms min) and because you could, if you wanted, switch it to 8 Ohms and still add another cab to it later.
As for mixing 12s and 15s, that's not something I'd normally advise, but the AK and Subway cabs are known to work well that way. (Note, if using the 12" and 15" Hathor cabs together you would, in most cases, want to set the 1533 to 8 Ohms -- power handling is similar between the 12" and 15" cabs.) Anyway, I get if the 15" isn't your thing.
Re. efficiency specs, I think that they can be useful for comparing speakers within a line (assuming the manufacturer uses the same metric for all of their speakers), but, unless you really know what you're looking for (drivers used and other relevant parameters and/or how the manufacturer derived their specs) I'd take them with a grain of salt when comparing cabs between different manufacturers. My Markbass CMD121P combo, for instance, is listed at 99 dB efficiency. My Hathor 1203 is listed at 94 dB. I've played the head through the combo and the 1203 separately and with the two stacked. There is no way in hell that the combo is 5 dB more efficient than the 1203 (and I've given my opinion on their relative merits and the relative merits of the NY121 and Hathor cabs, a few posts up). There are different ways that this number can be arrived at and manufacturers are likely to pick one that is favorable to them.
Everything is relative -- the sound you're going for, room, and people you're playing with can all affect how much and what sort of rig you need. Among modern heads, I'd consider the LMIII to be moderately powerful, but not in the top-tier of micros in that regard. For single cabs in the size/form/schlep range you seem to be looking at, I like the 4 Ohm MAS 112 with the custom Eminence driver and the Hathor 2x12 (also 4 Ohms). A little smaller (and deeper-voiced), the MAS 28 (to which you could add a 18). There's the Hathor 1533 (which I've not played but have a good feeling about based on my experience of the 1203) and two Hathor 1203s (one or both of which could be a "tall" version).
The above would be my personal first picks -- but I repeat myself. There are lots of other cabs that
could net you a good result -- people have spoken highly of the Subway and Genzler cabs and the GK Neo 12s (I am not a fan of their Neo 10s -- at least not the Neo 410 cab or 210 combo I tried -- voicing was scooped for my taste), used Bergs, certain Ampegs, the Markbass NY 121s or 121H cabs, used Glockenklang cabs (a friend plays a stack of Glock 12s with a Markbass head), a stack of EA Wizzy 12s... there are others. Something with a fairly balanced response, good mids, and at least moderately efficient is what I would look for.
Good luck!