You keep using that word*. I do not think it means what you think it means.
Your assumption that Strahd's weight would be unchanged seems to come from a misunderstanding of the text you quote, "His statistics...are unchanged."
You are interpreting "statistics" with the common English meaning of 'measurements of characteristics'. However, "statistics" has a very specific meaning in 5e.
A monster's statistics, sometimes referred to as its stat block, provide the essential information that you need to run the monster. (MM p. 6)
In 5e, "statistics" are the characteristics, and only the characteristics, listed in the stat block (and see: What all is included in a creature's game statistics?). Thus, when CoS says that Strahd's statistics do not change, it means that everything listed in his stat block (other than size or speed) does not change in his transition to bat form. It makes no claim about characteristics not listed in his stat block.
Weight, crucially, is not part of a monster's 5e statistics. The passage you cite as evidence that Strahd's weight does not change is not, in fact, saying anything about whether or not his weight changes.
It is entirely the DM's decision then what Strahd weighs, both in his vampire form and his bat form. Perhaps he weighs as much as a normal bat in both!
*Yes, you used it only once. But my title is the full Princess Bride quote.