Yes, it is possible. I wouldn't recommend what you saw on Fiverr, because you have no guarantee that the developer will be available if you want a change after a certain time. And in the crypto industry, things change very quickly, and regular updates are almost mandatory.
There are exchanges that, in addition to their APIs, also offer ready-made solutions for resellers; maybe you can start from there. They offer the whole engine, liquidity management, and transaction processing. It is yours only to maintain the site, hosting, and promote your exchange. At least 90% of instant exchanges today work that method.
Here's an example you can start from. (This is not my guarantee of service, but they've been here on the forum for a while, so they seem legit)
The Change Ltd
Their ANN on this forum:
https://bitcointalk.org/index.php?topic=5505862.0About the reseller partnership program:
https://thechange.ltd/en/white-label