If you cannot disable counter you have to add one AND gate and RC to delay 1 on the "second pin of the AND gate" (the first pin is from your uC, output go to counter).
If you can disable counter, then add a RC that will disable counter until the uC is up and running.
With regards to "without adding hardware" the only possible solution is that you set the initial pin state to input or 0 in config ... I do not get how come they go high, I do not know any uC that on power set all pin's to output and high ... most of them set all pin's to input, and if some are set as output, those are set to 0 and not 1 ...
From what you wrote, I'd say that your uC is powered on, it start the main program, then the RC circuit reboot it and then uC start program again from scratch. This brings me to conclusion that those "set as output high" is actually done in your application and not by uC design... if that's true, all you need to do is be aware of the reset and solve the problem either by
"check the start registers (however is that done with that uC) to see if you were rebooted by RC or POR" or "at the beginning of your application add a delay that is longer then RC time to reset the uC... It will increase the bootup time but will solve you problem in software ...
The first solution is better (detect if you just started or you were reseted by RC)
Sorry that I cannot be of more help but I never used that uC