Allow me to explain why this is XY problem.
You've come with a solution Y to use RSTn signal to trigger 555 timer. This created a problem that you've asked to fix:
How can I neglect this initial startup pulse to avoid triggering 555 timer during initial startup?
The actual problem X, however, is:
"How do I trigger 555 timer periodically using TPL5010?".
Had you asked this question, the answer would have been: TPL5010 is designed to generate WAKE pulses periodically, and that is what you should be using, not RSTn. Note that in order for next WAKE to be generated, TPL needs DONE pulse coming in. In circuit without microcontroller you can use the output of your 555 timer to generate DONE.
Now, the funny thing is that your problem is not even XY problem, it is XY² problem. Because the real question should have been:
"How do I generate 10 sec pulse every N seconds?"
Then the answer would be completely different. First, the timers TPL5110 and TPL5111 are much better fit for this task. They do not require DONE signal to repeat the output pulse. Instead they wait almost to the end of programmed period and then toggle output back anyway. At the same time DONE can be used to de-assert output earlier. By looping back DRVn output to DONE input via RC delay circuit one can control the duration of the output pulse.
See, what I am getting at? There is no need in additional 555 timer. It all can be done with just a few parts. But you need to ask right question to get this answer. In the circuit below R1 sets the pulse period, R2 and C1 set the pulse duration.
BTW, you can try the same approach with TPL5010, by looping delayed WAKE pulse. Although I suspect it wouldn't work if the chip waits for first DONE to begin time interval.