Does the Biba integrity model (No write-up, no read-down) necessarily conflict with the Bell-Lapadula security model (No write-down, no read-up)? Or can they both coexist in a single system?
They don't absolutely conflict, but it puts severe limitations on access. With both of these at the same time you can operate on the horizontal or a per issue access...but this is a pain to implement. I'm not sure why you are asking. The models are typically not combined in application.
It would be the equivalent of having every communication as eyes only with a notes section at the bottom, and you only hand it off to specifically authorized people...does that make sense?
I have seen an environment set up to effectively work like this, but it was for a specific reason. It was an airgapped network with access only for 6 people who had an identical role. All data in the network was there from the beginning of the project until the end, and then the results were reclassified.
Don't think you'd want to do this in real life, as typically you do want information to move from one area to another.
For questions like this I would suggest posting on https://security.stackexchange.com/ as the range of experienced folks over there could provide a few viewpoints.