If you already have lightning tip bot setup then you can just open the other wallet (the one you want to get sats out of), tap “Send” and then in the invoice field type your LN.tips Lightning address (see screenshot below as an example). Tap “Done” and then on the next screen make sure you’re sending from the correct wallet (bottom of screen) and enter an amount lower than your total balance, since BW has that minimum reserve requirement for routing fees. Then complete sending/confirming the payment.

If BW is charging higher fees then there’s nothing you can do about that unfortunately. Hopefully that’s not the case.