Ah yes an
ECV is out, safety before all.
Still wondering how smart or not it can turn out to be not to have your own chair within reach during the travel. You would not be the first one that are confrontated with their VS acting up while at the airport. If it does, you would also not be the first one denied boarding on said situation if there remains even the smallest doubt that you might not do perfectly well in air. Not trying to rain on your parade or to be a Debby Downer, but having seen it happen before I do want to mention it.
As far as self propelling is; be prepaired to use more help than at home and perhaps expecting. Not just the heat, but the sheer fact that you are looking into using a rental. Those will always require more of the user and/or those pushing. Even more so if you are using a sports chair normally.
As far as having a connecting flight; that is no biggie AT ALL. You can basically decide what happens. You can have a chair labeled through, in where it would not come to you at NewArk/JFK but get loaded right onto the flight to MCO where you will see it again. Also you can decide to have the chair brought up to you. Simply deplane last and 9/10 times it'll already be waiting for you at the gate. Simply use it to go through CPB -be forwarned; lines and waits can be long and sometimes these halls can have a warmer environnement-, spend your time in between flights and recheck it at the gate where they will simply load it again after you have boarded and it will again be brought to you at MCO. Honestly, no biggie at all.
I can understand wanting to be carefull with a chair (fly with a powerchair and manual chair with a combined worth of $40,000- myself amongst others), but there is one thing that would top that IMHO; yourself and your health. Chairs and any other aid can be insured if desired and damages that are minor in $$$ will even totally be covered by the airlines responsibility. If a damage does happen it sucks, but most of the times it can easily be taken care of and the end result tends to be more then worth it.
If you do decide to go down the rental route; be aware of rentals always being a step (or more) down from what you are used to. It will have an impact. Some things can be limited when taking precautions, others can't. One of the common problems; fabrics used. Being rentals they will have to be able to take a lot of usage, abuse, accidents and what not more. So fabric that fits the bill. Nice and peachy if you spill your drink on it

but a too easy real pain if not prepared. These fabrics get hot easily, get sticky, can literally rip skin open or cause rub burns. And they will make you sweat like no other. Yugh.

Limit those things by always having a cover up handy like a towel or any other smooth piece of cloth that breaths and takes up liquids like sweat and what not. Depending on type of rental chair it might or might not work to bring along your own cushion.
Communicate as much details with the rental company beforehand as you can. Find out sizes, options etc. etc. What is the norm for you might very well not be the norm in rental world. Best to know beforehand what is possible and what not so you can make an informed decision, take precautions if needed etc. Especially seating sizes are very important. Consult with multiple rental companies to make sure you find the best option for you. Also look into things that tend to be forgotten but might be important for you; for instance what type of breaks are on the chair? When being used to type A but finding it difficult to work with type B it is so nice to invest the few minutes avoiding the type B break where possible. And since you might need a push; also look into those options. What type of push handles are on the chair? Fixed, removable, fixed height or changable, those kind of things. Again when used to a sportsmodel, changes can be fast. Having this info beforehand can make the difference. Even if it is just knowing beforehand for instance that it might not fit into that rental car you were looking into for part of the trip.