Another advantage of a frigate is mast height, they will see the sloop long before the sloop will detect the frigate and they will investigate, to prevent a surprise by an enemy fleet, since many times a frigate was used as a screening force for the main fleet, given that they were as fast as a sloop, but better armed.
Again, why is anyone thinking that the captain of the frigate , is going to let any ship, sloop or otherwise, sail right up his stern, Frigate captains were usually very experienced, even daring and quite often inspired commanders and a Frigate was what everyone wanted, they ranged ahead of the fleet and were well suited to taking prizes (capturing enemy vessels or merchant men) and unless there was another ship insight, the captain got the majority of the prize money when the ship was bought into the service (a percentage went to the admiral in charge of his squadron) and the crew also got a percentage, you take this personality and a Frigate and you would be very foolish, to assume that he is going to let anyone get in position on him.
The Captain of a sloop is generally a junior officer and it may very well be his first command, it's the perfect platform on which someone can get the grasp of what it takes to command a fighting ship, without risking or impeding the performance of a vastly superior and more efficient fighting platform and being young and relatively inexperienced, he is already behind the curve when it comes to the tricks and deviousness, that a more senior experienced captain will most likely have at his disposal, add to it that his crew will have some experienced sailors, but will most likely comprise of equally fresh crew, ships of the line and frigates, were known to requisition, the more experienced men from lesser ships into thier crew's..
Basically the mast head watch detects the Sloop, the captain or indeed admiral of the squadron, should he be attached to one, will order that they begin to alter course to close and put him in an advantageous position, to first determine the what and the who and to allow him to run down this ship and either put it on the bottom or capture it.
As far as tricks, there were many tricks pulled by both sides in war, thing's like reinforcing a section of the ship and mounting a 24 pounder, removing the gun on the opposite side, to allow room for recoil, or replacing a 9 pounder bow chaser with a set of 12 pounders, even so far as mounting a pair of 24 pounder stern chasers, in place of a couple of carronades and since, it may not be policy or in the original design spec's, it could be an very nasty suprise.
Morale/psychology effect on the crew: A new weapon may indeed have an effect on the frigate crew, but I pose one better what is the effect on the Sloop's crew when this (in the case of the USS Constitution) 2200 ton behemoth with guns run out is headed for them, and every man of thier crew knows even a moment under those gun's is going to spell death (if they are lucky) and destruction and if any had served in a line ship or frigate, they know exactly what is coming, plus with a 50 cal, you won't have the immediate and physical feedback as would be caused by a cannon ball, so this too would have an effect on the crew of the sloop.
It works both ways.
So, if A frigate were to allow a sloop to close on his stern, then I for one would not want to be there, because the captain of that frigate, wants you there and given that he can quickly tack around to bring his gun's to bear and you at a 1000 yards, you turn either direction and he can place his broadside across your length, keep coming and he is sending cannon balls into the bow of your ship, then he tacks again and the other full loaded battery has a go at you.
It is not a static battle and since he knows it is a sloop he is after, the jockeying that would normally happen, with 2 ships of equal fighting ability, would not be present, he knows you are in serious trouble, should he get you into the firing arc of any of his gun's and he is going to come after you with all he has.
Money still on the Frigate