Provided the dogs already have EU pet passports, quarantine on return is not required. Dogs are also required to have had an approved tapeworm treatment between one and five days prior to the return date, with evidence from an official vet who administered it.
See
https://www.gov.uk/take-pet-abroad/pet-passport
"Third-country official veterinary certificate
To enter or return to the EU from listed or unlisted countries you need either:
a third-country official veterinary certificate and any other documents listed on it
a pet passport if your pet was given it before leaving the EU
...
You do not need a third-country official veterinary certificate if both:
your pet was issued with a passport in an EU country before it left
the treatments are still valid
Any booster vaccinations or blood tests carried out from outside the EU must be recorded on a third-country official veterinary certificate."
The US has its own rules about dogs leaving the country
Details are here:
https://www.aphis.usda.gov/aphis/pet-travel. Basically similar rules as returning to UK from EU on a EU passport. USDA or APHIS registered Vet inspection to be fit for travel, administration of approved wormer by them between 1 and 5 days before date of return to Blighty, together with some paperwork, in original ink, with an embossed seal. Plus Vet fees and administration fees/tax.
Not sure what happens if e.g. a return cruise is going to last over 5 days.
Also worth researching what nasties could infect your dogs over there, to which they may have no natural immunity, and taking sensible precautions.
Bringing pets into Great Britain: pet passports, Great Britain pet health certificates, microchipping, rabies vaccinations, travelling with assistance dogs.
www.gov.uk
Don't forget your International Driving Permit, required in some states.
Apply for an International Driving Permit from the Post Office for just £5.50 for the whole year! Find out more about driving abroad online here.
www.postoffice.co.uk
You'll need a Visa, since you'll be there for more than 90 days.
Then there is the issue of using an EU van with 230V 50 Hz electrics in a 110V 60 Hz country. You may need a transformer to convert the voltage (but not the frequency) if e.g. your
battery charger isn't dual voltage.
Something like this might work. NB it is designed for 50 Hz, "portable power tool" rated at 3kW, so not continuously at that level.
2x 13A Outlets Robust GRP Casing High Quality Internal Windings Carry Handle 2m Cable Length
www.toolstoday.co.uk
I expect there are heavier duty transformers available, designed for the job, this was just the first that I found at a reasonable price. It will probably handle 60 Hz OK. Even US 110V plugs and sockets don't inter-mate with UK yellow 110V ones, so you'd have to change that or make up an adaptor. At least take a yellow socket so you could fit it to the end of a US extension lead, which have to be heavier wire since they draw more than twice the current CF. 230v for the same power. I.e. just converting a UK 16A cable would give you less than 7.7 A equivalent rating, that's just 1760 Watts.
Also different gas systems. E.g a UK POL propane pigtail will not screw into a US gas bottle. Good luck finding e.g. a US POL to W20 pigtail. Maybe even fit a US regulator, but they run their propane at 27 milliBar, which is rather low for our appliances.
Other shippers are also available to deliver your van, worth shopping around.
It sounds like a fantastic trip. Friends of mine hired an RV, spent a month there and loved every minute.