The subject of a sentence can never be within a prepositional phrase (in a 'for' clause).
Your tenses also don't sound natural. To invite someone to something and to prompt something are straightforward tasks and I would say they demand the perfect past tense in this context, not the present perfect progressive. Thus,
For us to have been invited to the party prompted in us an inordinate pride.
To have been invited to the party prompted in us an inordinate pride.
The meaning is the same, but 'for us' is not part of the subject. I'll also comment that neither phrase sounds very natural/native. Infinitives are most often used as subjects in habitual/general circumstances (e.g. To study Russian is very difficult). So you could naturally say something like 'To be invited prompts an inordinate pride', but it doesn't sound quite right when you personalize it in the past. If you wanted to use a similar structure that sounds more natural, I would use a gerund (verbal noun), e.g.
Being invited to the party prompted in us an inordinate pride.