So after more than a week of troubleshooting every option I could think of and was advised to pursue, calling both Dell and Apple tech support, even nearing the options of a total system wipe after trying repair points, going to an Apple store in person and bringing both my old and new laptops with me along with my Beats headphones and several video recordings just to provide better visualization of my problem, I was able to scrounge up old Bluetooth devices to test on my new laptop, a mouse and a radio which both worked fine, isolated the problem to my Studio3 Wireless Headphones, and both Dell and Apple told me that, "I did nothing wrong on my end and this is likely a manufacturing problem with the Bluetooth chipset compatibility in the Dell laptops, older headphones will likely not work with the Dell Inspiron 16 Plus". And per both their advice, I had no choice but to spend $400 on a newer Beats headphones, Studio Pro, just to get basic audio functionality through Bluetooth.
Now granted I still hear my new Beats Studio Pro disconnect sometimes, it isn't nearly as often as with my Studio3 Wireless headphones, it's actually far rarer, they don't buffer the video at all, and they reconnect immediately and automatically. I guess, on some level, I can actually work with them on my new laptop with out too much of a hassle.
In any case, this entire troubleshooting experience has been thoroughly unpleasant and I'm incredibly disappointed at the laptop manufacturers for this excruciatingly infuriating oversight.