We had a mechanic look at the brakes
shows the [have + noun phrase + bare infinitive] construction, where 'have' is causative and 'had a mechanic look at' paraphrases to 'got a mechanic to look at', so 'have' has an 'engineer / implement / arrange for something to happen' sense.
However, in
We had a strange woman come to the door selling pictures
which shows the same [have + noun phrase + bare infinitive] surface construction, there is no arranging of the event by us. 'Have' here is non-causative, and 'We had" is a sentence-introducing element meaning basically 'This happened to us: a strange ...'.
One could say that 'have' here is, as in 'We had a nice holiday', a quasi-possessive (it's part of our history), but it's probably best to see 'We had ...' as used here merely as a short sentence-introducer, fronting 'we' before the main event, with 'have' almost a bleached-of-meaning function word (compare a in French Il y a).
It is unidiomatic to use 'We were having a strange woman come to the door selling pictures'. Though as BoldBen points out,
We were having a woman ring us as a wrong number so often that we
blacklisted her number
and (after Greybeard)
We were having strange men and women come to the door selling pictures
both use the 'experiencing', definitely non-causitive sense of 'have', are certainly acceptable. It is the plausibility of the repeated occurrence that governs idiomaticity.
Sorry, no supporting references.