Student in need of help: How to measure unidimensionality of binary MNAR data

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So for my thesis I need my data to be unidimensional. I want to test the unidimensionality using CFA. However, my data has some issues that make a standard CFA difficult, as it is MNAR and binary. So then how do I:

Pre-process the missing data? I've heard using multiple imputation in MICE is adequate, is this correct? And after Pre-processing, do I then use Lavaan for the actual CFA?

Estimate? MLSMV looks to be the most promising. Can I also use ULS, DWLS or WLS, why/why not? Or is there a whole other way that I haven't thought about?

If I've removed some data-points in the pre-processing, do they need to stay removed for the actual statistical analysis I plan to do after the test for unidimensionality?

Ziegler, Matthias & Hagemann, Dirk. (2015). Testing the Unidimensionality of Items. European Journal of Psychological Assessment. 31. 231-237. 10.1027/1015-5759/a000309.

Rogers, P. Best practices for your confirmatory factor analysis: A JASP and lavaan tutorial. Behav Res 56, 6634–6654 (2024). Best practices for your confirmatory factor analysis: A JASP and lavaan tutorial | Behavior Research Methods

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The forum is essentially an R forum. There are extremely good statisticians here but I think you are asking a complicated enough stats question that you probably need to talk to a full-time consulting statistician, probably in person. There may be a good chance your department has a stats guru who is familiar with your type of problem.

Does your department or institution offer stats consulting? If so, I'd also try there. Or try the Math/Stats department.