I am putting in a new bathroom fan/vent. No fan currently. Since I have demo'd the shower insert to put in a new shower I am also adding light and fan. The situation is as follows:

(1) The ceiling is vaulted 12+' at apex.

(2) The ceiling is beam and wood, tongue and groove 1" planks. Then 1" foam/poly insulation/vapor layer, and roofing material (composite shingles). In all ~3" thick.

(3) There is no power/conduit in/through ceiling as there are no voids.

(4) The only power and switch location is on an interior wall; the wall where the shower plumbing is located.

(5) I have placed switches and run power up that wall off a GFCI circuit for light and fan.

(6) I have purchased a Panasonic FV-0510VS1 "WhisperValueDC" ventilating fan as it is thin enough to mount within a standard 2x4 stud wall and rated for wall installations.

(7) The fan has a 4" oval conduit mount, for which I have a 4" to 3" rigid adapter - and 3" rigid ducting.

(8) I plan to mount the fan approx 12" below the roof line in the wall which is approx 2' below the roof apex.

The challenge is that I am unsure of what roof cap to use and how to best attach the ducting given (a) I cannot holesaw a 4"+ hole as that would exceed the diameter of the wall and placement (3 3/2") within it; (b) the thin ceiling/roof combination does not allow for any wiggle room for a 3" to 4" adapter required to fit most standard 4" vent cap systems; and (c) there does not seem to be a roof cap vent for 3" ducting other than the type for pvc/plumbing vent stacks etc.

Am I missing something? Any ideas on caps or other solutions are appreciated. The other option is to abandon the fan entirely. There was no fan previously which was not a big problem as the bathroom is huge with high ceilings, but the shower was also enclosed in a full shower stall which I am now opening up to above, so in theory there will be more steam moving out of the stall toward the ceiling instead of condensing on the interior of the shower.