Visa requirements for leaving Israel, going to Jordan, returning to Israel again

I am a UK citizen heading to Israel and Jordan in April. My concern is to do with border crossings / visa stuff.

Here is my plan:

1. Arrive in Tel Aviv from UK (visa on arrival)
2. Enter Jordan via The Jordan River Crossing (Nazareth to Amman)
3. Travel around Jordan for 4 days
4. Re-enter Israel (preferably via Allenby Bridge / King Hussein Bridge)
5. Travel around Israel
6. Fly back to UK from Israel

My questions are:

• Can I do step number 4: can I re-enter Israel after leaving for just 4 days?

• If yes, do I need to purchase a new visa?

• If yes, can I re-enter via the Allenby Bridge?

I can't quite find the answer to this anywhere (even after much forum searching and on here!) It seems like the situation is pretty volatile and subject to sudden changes.

This similar question here is about going the other direction (Jordan --> Israel --> Jordan). Jordan double border crossing?