I think the Pats likely borrow from what the Broncos do with their CBs:
LCB: Gilmore
RCB: Butler
LCB: Gilmore
RCB: Rowe
Slot: Butler
Butler will follow Landry, Sanders, Brown & Gilmore will follow Parker, D. Thomas, M. Bryant.
The are a couple problems with your projected alignment ...
1) Gilmore has played RCB his entire career and Butler has played LCB his entire career. Swapping them may seem trivial on paper, but it's not in the real world. It's not all that often that a player who excels at LCB is also able to master the reverse footwork and body positioning that's required to play the other side equally as well (and vice versa with RCBs).
2) Butler's exceptional coverage ability derives from his highly developed ability to read plays and anticipate routes, rather than from his natural athleticism. For playing the slot, natural agility and quick CoD are generally more helpful for coverage than anticipation.
BUTLER: 5096/187
4.62/40 .. 1.62/10-yd .. 33.5" vert .. 118" long jump .. 4.27 shuttle .. 7.20 3-cone
ROWE: 6006/205
4.45/40 .. 1.56/10-yd .. 39" vert .. 125" long jump ..
3.97 shuttle .. 6.70 3-cone
GILMORE: 6004/190
4.40/40 .. 1.52/10-yd .. 36" vert .. 123" long jump ..
3.94 shuttle .. 6.61 3-cone
RYAN: 5111/191
4.56/49 .. 1.58/10-yd .. 32.5" vert .. 116" long jump ..
4.06 shuttle .. 6.69 3-cone
Gilmore has dropped down to cover the slot for a few plays or a series regularly during his career with the Bills.
Rowe played nickel/slot as a rookie for the Eagles in 2015. With the Pats last season, Ryan began the year as the starting RCB with Coleman as the primary nickel/slot CB sometimes subbing at RCB for Ryan, with Ryan then covering the slot). Once Rowe was acquired, the Pats tried a lot of different combinations of Rowe, Ryan, Coleman and the Jones boys working RCB and the slot and appeared to settle on Rowe at RCB with Ryan as the nickel/slot guy.
Last season, Butler was used in the slot for a fair amount of snaps over a couple of games and gave up a QB rating of 128.
So, my guess is that the base starting combo is most likely to be ...
LCB: Butler
RCB: Gilmore
Slot/Nickel CB: some combination of Rowe/Gilmore, varying with the game situation
NOTE: At least hypothetically, Rowe has the size to play nickel "safety" in the "Big Nickel" package. It'll be interesting to see what, if any, experimentation goes on with that in Camp/pre-season games.
Cyrus Jones (6.71 3-cone) also may be in the RCB/nickel/slot mix or possibly one of the UDFAs. There may also be some adjustments and/or more overlap with the "Big Nickel" and dime package snap-count distributions, as well as substitutions at safety in general.
IOW, I think there's likely to be a fair amount of experimentation, even extending into the regular season, to determine what combinations of five and six DBs works best in which situations. However, I strongly doubt that any of that involves Butler playing any position other than LCB.