In the case of parve food cooked in a clean meat pot, there is no need to wait before eating dairy and no need to say birkat hamazon. Indeed should the parve food become mixed together with milk, you could even eat them together after the fact.
R Yitzchak Yaakov Fuchs describes it in his book Hakashrut (p. 373)
One may eat dairy foods immediately after eating parve food that was
cooked in a meaty pot even if the pot was not thoroughly cleansed from
the greasiness of the meat that was cooked earlier in it.
(YD 89:3 with Schach 19. See also here for a summary, and here for the detailed laws. Please note that this doesn't mean you can mix the chili together with dairy.)
Saying birkat hamazon might in fact create a problematic situation of unnecessary blessings. This is because you knew before saying birkat hamazon that you still wanted to eat more.
In this area, small details can make a big difference. So, consult your rabbi
before implementing anything you learn here.