Insurance goes through cycles like much of everything else. When I started flying twins (late 2008/early 2009) the market was not great, similar to what it is now. I was fortunate enough that my instructor had thousands of hours in the exact aircraft type (and several hundred in the exact specific airplane) I learned in. Then it went through a period where basically anyone could get insured in any upgrade, and people were going straight from an SR22 to a 421, which is a really bad idea. We're now back in a more stringent cycle, which is annoying for those transitioning upwards but it just is something you have to deal with.
Basically, your first year insurance wise and the first part of the transition will suck. After that, it does get better. They want multi time. Retract time helps as well, but multi time is the big thing. Something with no turbos or pressurization for your first plane does help lower premiums some (mostly a low hull value helps there, but that's harder these days), but you're a high premium the first year because you're a high risk. It's that second engine that's the part that adds risk because you have no experience with it when things go wrong, it's a lot less about the airframe itself.
But, it does get better. By the time I got to the MU-2, insurance was a non-event, but I had around 2500 hours of multi time including cabin class and turbine, so there was a good case that I'd make a successful transition. Really, every transition after the Aztec was pretty easy. The Navajo they wanted 50 hours of dual when I transitioned to that (despite having 1,000 hours of multi time by that point - that was in a particularly rough period of insurance), but when I bought the 414 they just wanted me to get an initial at an approved training facility and that was all. The funny part with that was that they let me do the ferry flight on a plane that'd been sitting for 6-10 years and only flown 100 hours this century (this was in 2016, so a lot of sitting) with zero training and zero 414 (or 400 series Twin Cessna) time at all.
The best thing you can do is fly the pants off of whatever twin you buy. It'll be good for your proficiency and then you'll have an easier time on year 2 and beyond.