Anova II table, df, drop1 and very complex regression models!

I'd probably use Mplus (www.statmodel.com) for that, [...]

[snip]

Fortunately, the lavaan package in R replicates some of what Mplus can do. 

Every couple of years, I mention OpenMx on this list:

http://openmx.psyc.virginia.edu/
https://cran.r-project.org/web/packages/OpenMx/index.html

I don't know exactly how much of Mplus's functionality it provides, but suspect it would be close to 100%.

As to the OP's question, we don't know how much data is actually missing, or what pattern that takes. We also don't know why some of the fitted models differ by 0 d.f.  If we did impute the data 5 times, wouldn't that give us an 80 h run-time?

Cheers, David.