Low resistance current limiting on GPIO pin
Why don't you drive it from some power line able to support higher current. For example, from Vcc=5V directly via P-ch MOSFET, that you will open and close using GPIO. I suggest some 10k-100k (you can do some math and experiment, but any should work probably) pulldown on that line to make sure the line is at 0 when the MOSFET is not conducting (pulldown on drain side on my picture; load on my picture = your line's output). If you find MOSFET quick enough, and there is plenty of fast MOSFETs, you'll be fine. After all, your intervals aren't all that quick, I'd bet almost any logic level MOSFET that works with 5V would fit. Provided it's rated for the maximum current you expect, not that it's big anyway.
CORRECTION
Since the receiving end has built-in pullup resistor, the transistor's role would be to pull the line down at the right time. For that purpose you use N-channel or BJT instead. A quick google search revealed that BJT are faster switches than MOSFETs, but these days they're all quick, and certainly quick enough for your application, so it's up to you (but don't forget to look at the datasheet). This correction is sponsored by @Justme.