We can make a tremendously good guess, though, because the French navy is so much more powerful. The USN simply does not have a battle line, and the French have enough battleships to form a fifteen-liner battle line.
The Paris treaty was an international one, not a bilateral one; the likely outcome is that the British announce that US privateers do not have the protection of their letters of marque. When added to all the other impediments to privateers (like how they could only run prizes into American ports, all of which would be blockaded, and the dearth of guns) the likely outcome is that privateers would be few in number.
US navy ships out cruising is the more likely form of raiding.
Your picture of 19th century naval operations is simplistic, and possibly informed primarily by US naval vessels.
Firstly, there is nothing inherently wrong with sailing across the Atlantic largely with sail, but ships also made transit at higher power levels if need be. It happens that there was a coal mine on the North Atlantic station, so it was possible to burn coal on the way out and then replenish there.
Secondly, not all ironclads had to be towed (the Warrior is an obvious example of one that didn't have to be) but towing clearly worked - the
Terror did it, and if you look at lithographs of her she's kind of ship like!
Thirdly, the bearings wearing at an astonishing rate is specific to certain post-war US commerce raiders (the Wampanoag in particular). Here's some of the actual times that British vessels on the NA&WI station spent between refits:
HMS Agamemnon: May 1859 to October 1862
HMS St George: June 1860 to February 1864
HMS Edgar: May 1859 to July 1862
HMS Hero: March 1859 to November 1862
HMS Nile: March 1858 to April 1864
HMS Immortalite: November 1860 to July 1864
HMS Melpomene: June 1859 to March 1863
HMS Phaeton: November 1861 to March 1865
HMS Mersey: December 1861 to January 1866
HMS Orlando: March 1859 to August 1862
HMS Diadem: August 1857 to April 1862
HMS Ariadne: November 1859 to March 1864
HMS Cadmus: May 1859 to May 1863
HMS Challenger: May 1861 to February 1865
HMS Jason: November 1860 to December 1864
HMS Greyhound: December 1859 to November 1864
HMS Rinaldo: May 1861 to February 1865
HMS Peterel: February 1862 to August 1865
HMS Racer: June 1858 to December 1862
Fourthly, refits did not have to be very lengthy. Using samples from the Flying Squadrons which were unusually heavily used, sailing all around the world:
* Three frigates (Immortalite, Inconstant and Volage) entered refit on the 12th October 1871 and left on the 30th November 1871 - this is 79 days for the longest of the three (Inconstant) with the other two leaving for Spithead on the 10th November (59 days).
* The Immortalite paid off for refit on the 2nd October 1872 and left under sail on the 9th December (68 days).
* The Immortalite arrived at Spithead on the 7th January 1873 in order to repair a leak, and left the British Isles on the 16th-17th February (41 days).
* The Endymion took from 10th May to 12th June 1869 to refit; this is 33 days.
* The Barrosa (corvette) was ordered to fit out for sea on the 7th June 1869; despite an accident during preparations requiring a refit, she left on the 23rd June (16 days).
* The Narcisssus and Cadmus (corvette) took from the 14th October 1871 to the 13th November 1871 to be repaired; this is 30 days.
* Narcissus and Topaze took from th 30th September 1872 to the 12th December 1872 to be repaired and refitted; this is 73 days.
It is possible that those who argue the Royal Navy would take inordinately long times to refit their ships are basing their estimates on Union practice. The brand new screw sloop
USS Brooklyn, for instance, was out of service between 25 August 1863 and 14 April 1864, while the similarly new screw sloop
USS Seminole was decommissioned between June 1862 and June 1863. However, this overlooks one significant point: the number of dry docks. In fact, there were
only four naval dry docks: two permanent docks at New York and Boston, and two floating docks at Philadelphia and Portsmouth (with a third floating dock at San Francisco). This was sufficient for the tiny pre-war navy, but wholly inadequate for wartime requirements, resulting in a bottle-neck of dock capacity.
So putting this together, you have a situation where it's entirely plausible for a Royal Navy ship to spend a few weeks sailing across the Atlantic (at four knots to get to Halifax would take about 25 days), remain on station for a couple of years continuously (being resupplied with coal at the blockade station) and then go home for a refit lasting about two months before coming right out again. Not exactly the dire picture you've presented.