It sounds as if you may need to experiment around a little bit with string sets too see what works for you .
Over here ,what I do is spend a lot of my noodle and practice time playing acoustically on an Arch Top w 12's or 13's .
What that means is I'm most comfortable playing on heaver gauge strings ..over 9's or 10's
Most of my Solid Body guitars have 11's on them and for whatever odd reason it hasn't stopped me with my bending.
A recent song I did called My Good Tele Gone Bad is mostly bends on the dirty sections …guitar was strung w 11's
The Tele I'm playing tonight has 12's on it tuned to standard and I like the way it feels and sounds …
Regarding your situation , a good bet would be to pick up a set of 11's first and give that set a try .
If you are gonna tune down to D I doubt you will have to change much if any thing …intonation or truss rod ….
12 's would probably feel just a little bit tighter tuned down to D ..not by much over 10's ….feel wise …
Since I have not looked at your Epi , I do not know if you would have to adjust the rod or intonation …
I used 12's tuned to standard on my Epi Limited Edition 61 SG w P 90's for about 6 months and it didn't need a truss rod adjustment …now that guitar has 11's on it
A good rule of thumb is to take any changes you make on a guitar to guitar basis….
As far as the heavier gauge strings go …they will usually last a lot longer , also if your nut is not binding they will stay in tune a lot better ….
If you happen to be a guitarist that is looking for a Piano Bass type of sound on your lower strings or more punch
Heaver gauge is the ticket …
A couple of things to think about also …
When you start using a beefier gauge set of strings ( even tuned down ) you can really dial in your tone .
I do all my own work so I pay close attention to setting my truss rod correctly , the intonation on my bridge and setting my guitars action to ring out ….
Once having done all that correctly you are now in a position to set your guitars pickups to exactly the best tone for your playing and style
by adjusting the pickup heights …..
What happens is the pick ups are magnetic and they play off the vibrations of the string and the guitar body's resonance ..
( even solid bodies regardless of the wood )
When you have a beefier string , you have more metal vibrating in the pickups magnetic field ….
Most people that don't know what they are doing think that if you raise the pick up that the guitar will sound better
The guitar may sound louder and much muddier ..not better ….
To prove that to your self all you have to do is plug in your guitar to your DAW and do a real time read off your EQ 's.
When the pickup is too high it creates too much magnetic pull and it may inhibit the string from vibrating freely ..
In the case of a Fender Stratocaster , if you raise the pickups too high it will actually screw up your intonation
I have a few guitars over here with EMG's ..Those guitars have to have the pickups up high due to the low magnetic pull EMG's exhibit .
Anyway my point is if you decide to try a heavier set of strings even for down tuning , you may want to investigate on your own some of the things I have been saying here …
With the heavier strings you can lower your pick ups to the sweet spot and get a much better sound
more balanced more unique .
all the best,
Kenny