Funny how you identified all the viable options but limited yourself to getting the full set from only one method!
Your
easiest/fastest way to get your full exotic set would be to buy the pieces you can afford with karma with karma, the pieces you can afford with gold with gold, and save up dungeon tokens for the rest. On the TP, you can just search for "Solider's" armor directly.
There's no reason you need to get all 6 pieces from karma or all 6 from dungeons or all 6 from the TP.
Then you can, at whatever pace you please, obtain the skins you want and transmute everything to the look you want.
Luckily, SE1, SE3, HOTW1, and all three paths of AC (especially 1 and 2) are rather easy and fast. If you like the visual appearance of any of those sets, I'd head to them and get good at the runs- you'll earn oodles of gold in the process.
Another pro-tip: On the TP, do
not limit yourself to level 80 exotics. Search for level 75-79 exotics for massive savings over level 80s for a ludicrously minor stat difference (hopefully you're not the min-maxing type... this game doesn't much reward it

). As an added bonus, depending on what skin you want ultimately, you might be able to use regular transmutation stones on those pieces rather than Fine stones, which is always pleasant.
edit: Some relevant info/discrepancies...
Karma, besides dailies and monthlies, is earned from doing Events, Dungeons, Fractals that are below your personal Fractal level (help newbies!), and even Hearts.
Transmutation stones and Fine transmutation stones are earned from zone completion, rarely from completing your dailies/monthlies, and can be purchased in the gem store (of course, remember that you can buy anything in the gem store with gold).
edit again: Of course there's the matter of runes. If you have your mind set on a rune, consider looking for pieces on the TP that have it already, or choosing the dungeon/karma vendor that has the rune you want for the right pieces, etc. If you aren't worried about a rune set yet, then just focus on getting the pieces.
