This will probably too late for the OP to solve her issue, but because FT does a good job serving as information source when searching related info on the web it may help others, so here's a PSA:
Zinger makes semi-affordable crates for large dogs in sizes up to 32"W x 48"H x 60"L. (and larger for custom orders).: https://www.zingerwinger.com/Crates/Need/AirlineTravel/ . Those are the cheapest ones I know.
When adding shipping and the mandatory airline travel kit (https://www.zingerwinger.com/airline-travel-kit.html) the cost will still stay well below $2,000 even for the largest variants. They can be completely disassembled for easier storage and transport, takes about 20 minutes for two people with a screwdriver and a wrench.
Especially when moving from the US to Europe and this would be a one-time only use, it's absolutely no problem to resell them over there especially in Germany, Austria and Switzerland. The LH group - one of the largest players in the worldwide animal/pet transportation market - will not transport a number of dog breeds (those on their 'dangerous dog' list) in standard plastic travel crates but demands this kind of crate for passengers wishing to transport their dogs as checked baggage (see here for details). In Europe, that kind of crate is even harder to come by short of very expensive custom orders so there'll be demand.

AA has aircraft-dependent crate size restrictions. Live animals are only loaded into the forward cargo compartment (as only that one is climate-controlled on Embraer-, Bombardier-, Boeing- and Airbus A321T aircraft), so the size of that door effectively limits the crate size.
Back when AA accepted animals as checked baggage for the general public (which they did until 2020, a full decade after their primary competitors stopped to do so but unfortunately have stopped apparently for good now) one had to piece itineraries together by themselves using the fleet guide on aa.com and the schedule available on aacargo. if with a Giant or larger dog crate.
Now that this is history, such detailed knowledge isn't necessary anymore as one has to rely on the pros at AAcargo/PetEmbark to do their job. As a direct result, fees for transporting animals in the hold have gone up alarmingly. Where it once was an already-not-cheap $200 to take a dog from LA to NY on AA, they now charge more than $600. It's even worse internationally where four-figure pricing is the norm these days.
Today, the only pets-as-checked-baggage-to-Europe option remaining for those not living in a major city with no direct service by EU airlines allowing pets in the hold basically is Alaska for domestic flights (who will increase their checked pet prices by 50% come May1!) combined with an European airline still allowing them as checked baggage. Not ideal, as Alaska's route network is rather small compared with the big Three.

Cunard is indeed a good option, especially for dogs so large/heavy they won't fit a standard airline crate. It's cheaper than transporting them as cargo, too. Most airlines accepting pets as checked baggage for somewhat reasonable prices starting at $300 these days limit size and weight for this. Flying transatlantic with oneworld airlines, it's 45kg combined animal/crate weight with Iberia and 75kg with Finnair, Royal Air Maroc and Royal Jordanian for example - anything above that needs to be shipped as manifested cargo with stratospheric pricing.
The transatlantic trip on QM2 takes six days, not two weeks. It's travel a bit like in the old days before the jet age, and it's definitely travel, not your typical cruise. They get branded 'crossings' and not 'cruises' by Cunard, for a reason.
A large pet kennel is 36"x 36"x30″ and if that's too small two can be combined for a space that's 36"x36"x60". One kennel costs $1,000 (hooray for Cunard to not having raised their prices at all the last few years, they've been constant since at least 2018, something that definitely isn't true with airlines), Also, the animals basically just sleep in them and can roam around indoor and outdoor freely as they wish all other times.
Very true about the demand, the kennels tend to be booked solid a year and more in advance. But there's waitlists you can get added to, and when there's cancellations the slots open up again.
$1,000 for the pet plus a tad less than $2,000 for the accompanying human doesn't sound that bad compared to business class airfares plus pet fees these days.
Here's some good and up-to-date info about pets on the QM2: https://cruisemaven.com/travel-with-...ls-to-england/