Bridging Finance

Leave that option as a last resort. Talk to your broker or banker and see if you can service both debts taking into consideration the proposed rental income or the property you are not living in. Who is your lender?

I would agree that bridging finance can get messy. Trying to arrange same day settlements is not a thing you want to be involved in (which is ok because you mentioned its your PPOR in purely investment properties, not possible to have different settlment dates). You'll spend the extra costs in interest, fees, moving your moving dates several times if delays. It can cost just as much, although in theory sounds great that you settle on your house and move into the next.

in reality you should sell your existing house, have a offer go unconditional and then start looking. Otherwise you really don't know your budget.  Short stay accomadation is not that expensive and you'll have less headaches. 

Whether you get the new loan for the new purchase with the same bank or a new bank is a smaller part of the equation. You really should have an approval in place for the new loan with the assumption that you still have the existing loan debt. There should be no requirement for you to sell the current property. This is what you should be aiming for. 

St George has many short comings but they have pretty good servicing so it may work with them.

Most if not all lenders are now giving little privileges but these privileges don't mean much when their post settlement service is lousy to say the least. They also need to get real with their LMI premiums which are just uncompetitive.