You would just need a toggle Flip Flop, a one shot and two active low switches with Relay coil common to V+.
Single push button goes to
one shot which debounces push button and drives FF Clock input in /2 mode or toggle mode ( D=Qbar)
Q AND one shot = Gate drive 1
Qnot NOR one shot = Gate drive 2
FETs would be N Ch enhancement type.. most common. with RdsON<< Coil Rs.
AND function could be done in logic or inverted inputs with Diode OR and R pullup.
Make sense?
Here's proof concept.
Click on the switch bar.
Momentary switch should be gold plated signal type. (non-oxidizing low current)
FETs can be any RdsOn << Load Rs. Add reverse clamp diodes to coils.
This is a non-retriggerable one shot, leading edge of push button, and variable duration set by RC with Schottky signal diode to reset cap fast.
Scope traces show switch, Power in each coil (dummy 100R) and complementary toggle states.