Shotgun or dangerous game gun? I lean towards shotgun just as Jim Kelly, the gunsmith who worked on the gun, concluded. The 36" barrels seem to tilt towards that conclusion. The Reilly dangerous game guns both rifle and percussion smooth bore seem to have the 26" or shorter barrels.
(I will add that normally a Reilly gun with a pistol grip was originally a rifle. There are very few exceptions to this conclusion after looking at 600 Reilly's - and all of he exceptions are 8 bore or larger wild-fowlers).
EDIT: I have just gone over photos of 600 existing Reilly's from 1829 to 1912 and looked at all of the guns 10 bore or larger. In reviewing the photos, there is not a single certain instance of a Reilly shotgun of any bore using a pistol grip or a trigger guard extension which aped a pistol grip. There are about 6 large bore Reilly shotguns with pistol grips...I suspect all of them to have been originally rifles. Here are the extant Reilly 4 bores and 6 bores in my database (in addition to George L's gun); Invaluable Auctions will simply not let me ask a question - a pity:
SN
18860 - 1874
http://jamesdjulia.com/item/1363-396/4 bore, U-L hammer gun, wild-fowler shotgun. 42"? barrels. 18 lbs 12 oz
(Note the Safety)SN
15964 - 1869
https://www.gunsinternational.com/g...m-quality-condition.cfm?gun_id=1012452854 bore, Back-Action U-L hammer-gun, express rifle; 26" barrels, 18 lbs 10 oz.
SN
4573 - 1842? (JC - Outlier SN)
https://www.rockislandauction.com/detail/76/400/engraved-jc-reilly-percussion-sporting-gun6 bore percussion dangerous game gun (Owned by Terry Buffum)
SN
84xx - 1847
http://www.simpsonltd.com/product_info.php?products_id=425936 bore, percussion smooth-bore with 26" barrels.
SN
8578 - 1848
https://auctions.holtsauctioneers.c...320++++509+&refno=148046&image=06 bore percussion single-barrel wild fowler, 42" barrels
SN
7802 - 1853 - JC "7000" number series
http://промкаталог.рф/PublicDocuments/0523722.pdf
6 bore percussion wild-fowler - 42" barrels
SN
10315 - 1857
https://www.proxibid.com/aspr/REILLY-LONDON/22392022/LotDetail.asp?lid=22392022Percussion single barrel 6 bore wild fowler - 38" barrel
SN
11937 - 1861
https://www.bonhams.com/auctions/14918/lot/52/4 bore, center break U-L, hammer-gun (converted pin-fire) wild-fowler, 39" barrels; 15 lbs, 2 oz
Dave getting kicked and back-blasted by an 8 bore Reilly Rifle (note pistol grip - 26" barrel). (Note: Eyes closed? Hair flying. Is that pain? )