If the document is frequently updated, you may assume it is part of the working set residing in RAM. So as read is concerned, it is better to have it in the same document, otherwise the working set is bigger, since you now have 2 documents, with indexes, …, for the entity.
However, if it is very frequently update, it might be helpful to separate it. As I have learned recently, a document is completely written when it is updated. So it might be better to have it in a smaller dedicated document. I italicized might because performance is seldom intuitive. There are so many factors, that testing is always best.
That is why, it is better to make it work correctly and then fix performance problem, if any, after.
Something that might, yes in italic, help, is to have a change stream to monitor this value. This way you do not need to read it because you just have it.