Why can't it connect with the peers from the Bitcoin network? It doesn't have to receive any blocks just to send them a transaction.
In addition to the replies above, if you don't want to broadcast a transaction via Electrum, then once you have signed it you can simply export the raw transaction hex and then use a link such as one of the following to broadcast it instead:
https://mempool.space/tx/pushhttps://blockchair.com/broadcasthttps://blockstream.info/tx/pushhttps://coinb.in/#broadcast-snip-
You can do things such as alternate SPV servers each time you make a query to reduce the amount of information any single entity will collect about you.[/quote]
In addition to different servers, it would be wise to use a different Tor circuit for each query. Querying only a single address at a time is not the usual behavior of most users, and so would stand out from the crowd. The same IP address querying a single address each time would be fairly easy to link, especially since given the limited number of Electrum servers you will still end up using the same one repeatedly.
The bottom line is all of these methods are complicated, time consuming, and do not actually fix the problem.