Your description of the primitive resort in Fiji sounds like most of the places I've dived in South East Asia. I think you're probably asking a lot to get the large boats that you might get in the Caribbean in most of Asia. That being said, given the number of people who dive throughout SEA every day, I'd be surprised if you weren't diving off a purpose-built, or specially modified, boat the majority of the time. However, it's probably going to be based on the local model - so Bangkas in the philippines - because they're cheaper and more readily available (as are parts when the engine inevitably has an issue).
There are some photos on the Atlantis Website which suggest that they have a bangka which has been modified/purpose built for diving - i.e. tanks have specific tank stands up the back and there are seats on the side of the boats.
I highly recommend emailing the resort and asking them to send you photos of the dive boat, and explaining your situation and what you need in terms of diving support. They will let you know - I'm sure you won't be the only person who has been concerned about the boats. I know I often ask whether there's a ladder and how many rungs it has - if it's less than 3, I usually go looking for another boat!
If the entry is a problem, you could always get in the water with your fins, mask and snorkel and have your gear passed down to you so you can gear up in the water. With the water to help support you it might be lower strain on your back - just remember to put the BCD on first, and then the weightbelt!
In relation to shore dives, I know in Indonesia there were always people who were prepared to carry tanks to the water for a dollar, and then you could just float them out until you were deep enough to gear up easily. Would that be an option for you as well? I'm sure if shore-diving is an option, most dive operators will help make the diving fit your needs.
As another alternative, if time is not an issue, why not find a direct flight to somewhere else in South East Asia that you'd like to see (Hong Kong is probably a good choice because ), spend a couple of days there enjoying non-diving sites, and then take a short shuttle flight to the Philippines (Hong Kong to Manila is maybe 2 hours?).
Also, forgive me, I had to go look up a transom and couldn't find a satisfactory answer - I assume you are talking of a dive platform?
Anyway, I really hope your trip is starting to look more do-able.