QUESTION: Guitar is having a hard time staying in tune with my tremolo, and my e string tends to break frequently after long tremolo dives. How do I resolve this?

Hello, everyone, pictured is this beautiful Burny Les Paul with sustain pick ups that I purchased at the beginning of this year. I’ve finally purchased a Dusenberg Tremelo for the guitar, and while it looks and sounds great, I’m having a hard time of keeping this guitar in tune. When I do a dive with the tremolo, it comes out of tune. As a matter of a fact, when I use the tremolo for longer periods, not only do the strings come out out of tune, but the high E string will come out of tune, and if I try to re-tune, it will break. I’m looking for advice on how to solve this issue, preferably without drilling any holes into the guitar, or making any major alterations. Any advice is greatly appreciated! Thank you!