Term for faking or taking advantage of an incident to push an agenda?

A good description is "dirty trick".

= Underhand political or commercial activity designed to discredit an opponent.

Oxford lexico

If you insist on one word, three come to mind:

subterfuge = a trick or a dishonest way of achieving something

machinations = complicated and secret plans, esp. in obtaining or using power

ruse = a trick intended to deceive someone

[Cambridgedictionary]

Of these I prefer ruse, which is short and snappy and has a strong feeling of deception.

The suggestions above only apply to circumstances that were contrived by the beneficiary. If advantage is taken from circumstances that were not contrived it is opportunism.

Your specific example I would call a "false flag" operation, though I think it may not be broad enough to include actions not deliberately engineered by the person taking advantage of the situation for propaganda purposes.

It's not quite as simple as you describe. In order for this to work at any scale, an entire organization must be at the ready to take advantage of the situation. This entity is called Agitprop. The term originated in Soviet Russia as a shortened name for the Department for Agitation and Propaganda.

The second part are called war pretext incidents

I like ruse as @anton suggested. I would also add that the first action could have been a distraction, or to take that further, that the organization engaged in some sleight of hand the way a magician distracts the audience while he does something else. You might also say that the first action did not occur organically, but rather, it was contrived.