Well as for the unplugged sound, SGs tend to have thinner bodies and that changes the overall acoustic of the instrument, try confronting it with the Viper to see if this is the case.
As for the electronics, by a quick reasearch online I found out that your LTD's pickups have almost double the output level than the SG's and that makes a big difference in how sound is perceived.
Louder pickups tend to saturate your amp earlier, so you gain power but you lose a little bit of sensitivity and dynamic. There is no "better" solution, it all depends on what you want to play: if you like heavy genres try to reproduce the output of your Viper, if you want to play blues the SG is great as it is.

Different guitars are meant to sound different. If you want a backup guitar that sounds like your primary then get at least the same model/type or something that has the same pickups and wiring. If you're intending to buy guitars by the looks and expecting them to sound all the same, nobody can help you and nobody will because we all want a strat to sound like a strat and an SG like an SG. A world where every guitar in the world would sound exactly like some noob's only LTD that he likes so much because he never played anything else might be a plot for a small niche horror movie of very limited entertainment potential at best.

They will always sound different to some extent but the main tone change will be pickups (I'm guessing the LTD pickups will be higher gain). You might also want to change the other electronics (matching values there could help).

New strings will sound brighter but so will thinner strings. Try some different gagues and see what feels and sounds good to you.

More generally, does it need to sound the same or could you find a different sound that you like? Part of the fun of a second guitar is the variety and how different sounds and feels bring out different playing.

The Viper's pickups may just be hotter, human ears perceive louder == better. Try to match the output and see if there's a difference (there isn't).
As for unplugged, can't help you there; I don't really care much for how well they sound unplugged, since it's not really a use case

I mean if you really want to go all out, yes, you need to match string gauge, type, brand, put same pickups/electronics in both, etc.

But in my experience, I usually need a year to figure out a new piece of gear thoroughly enough to make any decisions. Take a photo of all your current settings with your current guitar and make some recordings . Then play with your new guitar and just keep trying to dial in the sound you want in all your gear for the next year .

After a year try comparing recordings or go back to your old guitar with settings from the photo and see if you still like it better. It's very likely that you will come up with a new sound that you like even better or arrive at roughly the same place you were before.

If not, you can swap out your pickups/electronics and strings and have it set up identically, etc. Just make sure the pickups are in exactly the same positions on both instruments because that will change the sound too. Wood type makes a difference but a very small one, and neck more than body.

A big part of getting new gear is rolling the dice and seeing what comes out of it. Not going through that process and just trying to force it to sound like something else is doing yourself a disservice. If nothing else, you will learn more about your gear and what you do/don't like in this process . But there's a very good chance you will come up with something new that you like. At least that's usually how it has worked out for me. This process has allowed me to hone in on the sound I need for the project I'm on and also be able to get a variety of different sounds for different things.