OP - Do not worry about bringing your meds into the US from the UK, even opiates and anti-depressants. You have been given some incorrect advice here; it won't be a problem. You don't need any kind of paperwork, including a physician's note.
And here we go.......... Might be me, but I'ld rather take the word of US gourverment for it than yours and guess what? Yes indeed, paperwork is involved in legally importing certain meds incl. opiats into the US. Not only been reconfirmed many times, but also have been asked for them, checked etc. It is upto Scojos to decide what she does or does not do and I couldn't care less about what someone thinks of me or being right or wrong, but this one of those reasons why I personally stay away from telling others what is and is not allowed because I am not the gouverment making up legislation nor the one carrying it out.
I travel internationally quite frequently with an opiate pain med (kidney stones), which I have taken to and from the US to the UK, Sweden, Finland, Italy, France, Hungary, Germany, Netherlands, etc. and not once has anyone given it a second glance.
Not being specifically checked does not make something legal. None of the countries have a 100% check rates for medication. And as always; each country differs, so have no validity to any other destination.
Scojos; tramadol being on the FDA-list means it is allowed onto the US-market and can be imported for own use legally. That is step one. Step 2 is always to check the CPB-info to see how you can import for own use legally. For some (types of) medications paperwork can be required. Nothing mindboggling, but still.
Before you pick the
ECV and rental care; think about your sitting for a moment. Are you already familiar with an ECV at home? I think not considering your trying out comment? Definately try out at home and if you can by all means go for trying out some foldable options.* It will even give you a better idea with how little you can "get away" without it becoming a source of pain or how much you might need. I know of those with FM that can happily drive around without any problem on the tiniest "foldable" ECV's, where others need a more sturdy option because if they don't get proper support from the seat and/or ECV at the right places they'll still run into getting too much pain too quickly. If you can try some options out -or perhaps already have loads of experience- you can reduce the risk of having an ECV that does not fit your needs or wants highly (depends a bit on how sensitive your body is on that subject or not). There most likely will be some differences between options in the UK and US (esp. when it comes to using rentals in the US and thus limiting the types somewhat) but the general likeness will give you enough info which ECV's are good options for you.
If it is foldable; great! Seeing your OCD-like-planning

I would then contact the rental company you want to use and ask for some dementions. Perhaps even compare those to info on the manufacturers webpage. With that you know how much space the car must have to make it doable. Then contact the car rental option and find out the options per rental category. Look up online what their trunk looks like and you know if you'll have plenty of space. Youtube is great for seeing with your own eyes how it works to take apart brandY-typeX ECV. Seeing always beats reading, if you ask me. With those few mails and surfing time combined, you have an amazing detailed answer of what will fit in where and how easy or nasty it might be to make things fit.
Peemag makes a valid point about the different type of crutches. Might be worth it to hunt down some crutch holders back home to avoid problems with the difference in crutches between both sides of the ocean. Besides; having one at home will be a help later on again whenever you find you might need to use an ECV at home. Even if it's once a year for a family outing, not needing to juggle crutches on the ECV is worth the small price for a crutch folder.
Are you considering any venturing offsite? If any ??? about that; shoot. I'm one of many on here that have done so with mobility aids in all kinds of ways. Public transportation, rental car, taxi, all kinds of options are represented on here. Personal opinion? Dump the men if need be, but go shopping!

Dollar isn't what it used to be, but still some great prices to be find. As far as accessibility etc. goes I can't say anything but great things. A must do for me. But then again, I'm a nut that already enjoys accesible public transportation.
Oh and let's not forget to tip you on bringing something to cover your seating and make you more comfy. In the Florida sun the fabric of rentals is hygienic but.... well......... sweaty, sticky on bare skin and gets HOT when having a peak of sun on it. Something as simple as a comfy towel can solve that. Whatever you find comfy. As long as it is something breathable that can absorb some fluid (sweating in the sun can be a bit worse when stuck to a seat or backrest on those areas), you will be loads more comfy than with the standard fabric. You are English, so no need to tell you about rain and seats getting wet when outside and you are not on it.
* tip; when buying your own, many companies will allow you trying out an ECV before deciding. Can you see it coming? Not saying go and buy one now unless needed, but it is an avenue to try out if the foldable type is not an option you could easily try out through rental, loan etc.