You can definitely take your handloads. You aren’t limited to factory ammo, but you are limited to 11 lbs maximum. Cast bullets would be a discussion with your PH before the hunt.
You’ll likely get a lot of offers sent privately but there are only a few sponsors I’d even consider for this kind of hunt. You are looking at $70k+ in any quality area, potentially a lot more depending on the type of elephant you’d like to hunt. I’d only consider offers from PHs based in the country you’d hunt with a few exceptions. If this is your first trip and potentially only trip and you want a safari of this magnitude, I’d reach out to the more established US booking agents like the hunting consortium, jack Atcheson and sons, hunters networks. They can present a lot of options and countries and discuss pros and cons of each for your situation. There are a lot of ways to go wrong if you don’t know exactly what you are looking for. If it’s before your surgery going to DSC or SCI show would provide you a huge amount of options to compare, but maybe too many.