Is there any easy way to make the counter count two steps for each button press?
Yes, I think so, but it's getting a bit more involved..
Use another XOR gate between the A & B outputs. The output of that will be high to indicate an intermediate step.
Add a set-reset latch with two cross-connected gates from a 4093, with the inputs from the switch signals to the counter. That will indicate the last step direction.
Make two schmitt R-C oscillators with two more 4093 sections, with the second input from the XOR output.
Connect diodes from the SR outputs to the oscillator caps, so a low level at the SR prevents the cap charging.
Add inverters to the outputs of the osc gates, so the signal is low when the osc is disabled.
Use the last two XOR gates in line with the counter inputs, after the SR inputs, and connect osc signals to the second inputs.
(I also forgot to mention in the previous one, the counter operation is rising edge so the count would occur when the switch is released. To have it work on press, you need a C-R differentiator plus a schmitt inverter after the debounce).
If that all works as I intend, given that I have not had my coffee yet this morning! --
Operate a switch, the counter input goes low then high - either switch press & release or the differentiator pulse when it's pressed.
The counter is then in an intermediate state, the output of the XOR between A & B goes high, whichever RC osc is not locked out by the diode from the SR latch starts to charge and after its delay the output from that via its inverter pulses high.
That inverts the output of the XOR in line with the appropriate counter input, adding a pulse & stepping the counter again.
As soon as the A & B match, that XOR output goes low and disables the oscillators again.
Four ICs? I've lost track of the number of inverters..
40106 for the debounce, differentiator and inverters.
74HC193 counter.
4093 for the SR latch and oscilators.
4070 for the XOR gates.