If I just want to send a BCH offline transaction (let's say to my new wallet), Can I broadcast it with an offline node?
You can:
1) Create the transaction with pre fork Bitcoin data (assuming you have air dropped BCH and never touched it).
2) Sign it however you want (offline included)
3) Use a BCH push service to broadcast from there (typically provided by block explorer websites).
Assuming magic bytes are only for finding nodes to connect to on the network (I believe that's so), and other hard fork changes in Bcash don't stop you spending in the old transaction formats (they've done something like 4 hard forks on the Bcash blockchain now, so who knows), then maybe alternatively:
3 b) Use your own Bcash software as your push service
This assumes you trust their software, and all the other assumptions at the start of this post.
The only advantage is that It could increase your privacy a little. Bear in mind that if you push a bunch of transactions to a website acting as BCH push service, then that website could easily collect some tasty data. If you push every output of BCH you own all that in the same browser session, it's reasonable for the website to assume that whoever pushed all the transactions also owns all the associated outputs being sent, and that's effectively deanonymising any corresponding outputs on the original Bitcoin blockchain (that also belong to you). Don't leak that information if you can avoid it!