normally, big NO, but I've read an article about Soloy in Alaska and they'd sponsor experienced vertical reference ops pilots with 500 time. I have no idea how OK would mostly 350 time be for them. Again, utility you're doing already.
Mostly Kiwi pilots other than US citizens who tick the boxTes and get there through word of mouth.
One pilot working for one of the HEMS companies mentioned there 'could' be option for green card, but I can't recall the handle/nickname nor specifics although they probably want twin and HEMS experience.
Not best time to get into offshore right now.
Many of us would drool over your logbook times/hours on type.
Hope you change jobs to your liking, but it seems that for some twin ops the companies prefer pilots with 'right type ratings' or IR(H) or not 'tons' of single pilot utility/charter/scenics. I find it silly, but I'm no them who make decisions.
More known user Vertical Freedom (and too zealous Robbie basher) not long ago had experienced indifference when looking for twin jobs and he's got plenty more time than you. It may not be picture of the European job market, though.