the inverter ic mentioned already has inputs on one side and outputs on the other - the datasheet even brags about it as a feature to ease board design.
however, I don't think you'll be able to just 'swap' a low side driver for a high side driver, their polarities are different. if your pcb was originally designed around the low side driver 2803, then the relay coils are already tied high and the driver completes the circuit, pulling the other side of the coil low.
ok, lets back up...
the problem is, when you hold your micro controller in reset, all it's outputs go high, and then turns on all your relays, which you don't want?
what you need is a 'bus driver' with an active-low output enable pin.
The 74HC541 should work for you, if you can figure out how to squeeze it onto your board. it is the non inverting version of the 74HC540 and has two active-low "enable" pins... when the enable pin is driven high, the outputs are floating tristate. when enable is driven low, by your uC after it boots up, the outputs become equivalent to the inputs (high/low)
as for fitting it onto your board, how about making a daughter board that will plug into the socket for the 2803. the daughter board will carry the '540 and the 2803. your idea of bending the pins should work too, plug the '540 into the input side of the socket and the 2803 into the output side, and solder the rest of the pins to each other