Has anyone successfully learned to read a dead language while skipping the formal grammar study?

Usually you learn any language by listening, speaking, reading, writing.

Also wanted to highlight that sanskrit is not a dead language. By dead language, I assume that it is nobody really uses it actively. I don't know about latin and other languages, but sanskrit is very much alive. The patronage decreased, but it seems to be making a comeback again.

Actually institutes like samskrita bharati, recommend newcomers to start with a 10 day-spoken Sanskrit class just to demonstrate bust the myth that one needs to learn grammer first.

Reference: https://samskritabharati.in/

An example session https://www.youtube.com/watch?v=d03uKp_OX18

Hope it helps.

For Latin, there is a book called "Lingua Latina: Per Se Illustrata" (abbreviated as LLPSI) which takes an interesting approach on this. I think this is an approach worth looking into, but there are not many books out there that do something similar:

LLPSI uses exclusively Latin from the first page, and the sentences become gradually more complex both grammatically and lexically. They repeat the same grammatical structure a few times with the same grammatical cases and verb conjugation, and add a tiny new element (Krashen's i+1) each time. A few people I know have gone through sections of this book and learned certain grammatical items from it without ever being explained the rules.

Note: I'm not advertising this book, just using it as an example of a method that might work well in OP's situation.