Checking bags all the way through on different tickets is complex but actually can usually be done if, and it’s a big if the agent involved is experienced and well trained. Sadly they are few and far between. It never hurts to ask however when you check in. Most agents simply don’t want to do it. I once had a agent tell me it was impossible to get two bags checked for me in a similar situation even when I handed her a printout on how to do it. I was able to get a supervisor who did it in about 90 seconds.