I would be pushing the operators pretty hard as an experience FO with good future prospects
I disagree. I am in the same boat as you although I have about 500 Pilot in Command time in a jet that I am no longer current in (total time is similar).
It will be pretty much impossible for you to get the direct entry captain position. It will be equally difficult to get an FO position. Unfortunately, experience is now often seen as a negative (shhh, don't tell the passengers this dirty little secret.) Your experience translates into being expensive. The operator can get a young, eager and rich cadet for a fraction of the cost.
I actually dis an experiment, I created several "dummy" CVs and sent them out to corporate jet operators and airlines. For example, for a RYR CV, I said I had 6500 hours total time (ATPL) and 1500 in the DC9. No response. I then sent a CV saying I had 373.6 hours total time with the Frozen ATPL and a "strong desire." Wooolaaah..I was contacted 48 hours later with an invitation to attend an assesment in order to become a cadet.
Next, I contacted a certain corporate jet operator in Germany and said I had 4075 hours total time, 2400 jet, and 400 in a Learjet 35. No reply, not only in Germany, but most of the EU. Then 2 days later, I sent a CV saying how I had 422 hours total time, but was more than willing to do what it took to get a flying position and what do know.... like magic, I received invitations for 3 assessments!
So, maybe you should redo your CV and not put down all that experience....it is hurting you. If you say you have 375 hours you are not lying...you do have 375 hours.
Good luck.