Is there any evidence of causation for breathing rate to affect blood pressure in a human?
Interesting question, it has been studied because sleep apnea is associated with hypertension in humans.
It's not my field of research, so I read a few sources to give you an answer.
There is a open-access review here (https://doi.org/10.1152/jappl.2001.90.4.1600) on the topic of episodic hypoxia, which is similar to the experiment you conducted. Two studies in dogs (https://doi.org/10.1152/jappl.1996.80.3.773 and https://www.jci.org/articles/view/119120) showed that neural reflexes, but also arousal, explain a part of the acute increase in blood pressure but here caution has to be used, as acutely holding breath may have different mechanisms on blood pressure elevation than the chronic hypoxia induced by sleep apnea.
Finally, a review (https://doi.org/10.1038/ajh.2009.43) in 2009 summarizes the evidence between sleep apnea and hypertension.

Overall, there seems to be multiple mechanisms. First, there is a sympathetic discharge during asphyxia, releasing catecholamines which can induce elevation in blood pressure by activating the baroreflex.  These are supported by studies of denervation in multiple organs. Secondly, there is activation of the renin-angiotensin-aldosterone system, a central player in blood pressure regulation. Thirdly, the animals studies show that arousal is an important factor in the acute blood pressure elevation ; in other words, being conscient of holding your breath itself induces blood pressure elevation. I cannot find the exact mechanism linking arousal to blood pressure elevation, though.