Hi there,
I can help a little:
"As far as I understand, we can't get married in the UK without her getting a special "family of a settled person" visa"
Sort of. The 'family of a settled person' visa is a settlement visa to live in the UK as the partner (or other dependant) of someone who is already settled there. Normally you apply for it after you get married, but engagement also qualifies as 'family'. There may be another visa that allows her to marry in the UK, but I don't think so. Maybe she has special options as a Commonwealth citizen? Can she check?
"Alternatively, if she was already in the UK, we could follow this path:
That is right. If she has been in the UK for more than six months, this would be the appropriate immigration route. I believe she can't apply for this on a tourist visa, but if she is working or studying then she could. You haven't given details of her status, but I infer that this is not the case?
"However, we would like to live in a place together earlier than that, and we were thinking of having a civil ceremony in the UK prior to the religious wedding in Italy. This would also give my fiance right to work in the UK."
I'm afraid not - merely being married to a settled person (or, indeed, to a citizen) does not confer the right to live or work in the UK. Having a wedding in the UK is not requisite - as long as your marriage in Italy is legally valid there, it will be legally valid in the UK.
Meanwhile, the EEA family permit is not a settlement visa and is only valid for six months. I don't think you can get married in the UK on it.
So to live together in the UK as soon as possible, your fiancee should apply for the 'family of a settled person' visa now as an unmarried partner (). You would need to prove that your relationship is 'genuine and subsisting', and especially that you have remained in contact while living in separate countries (if applicable). You would also need to demonstrate that you have a salary of at least £18,600 per annum, or savings of at least £64,000 (I think), with at least six months' history. You also need to have a suitable place to live with lined up, and she will need to submit a criminal record certificate and pass other background checks. This can take a few months from start to finish, so you may not complete the process long before your wedding in August.
Alternatively I guess she could enter as a tourist and live here without working until after your wedding, and apply then. Or you could have a quick civil ceremony in Italy and apply as a married couple. In any case, you're ultimately applying for the same visa.
However, I strongly recommend that you not take my word for any of this, nor the word of anyone on this forum. Instead, I recommend you consult an immigration lawyer. This will add to the cost, but it will be worth it - they will have comprehensive, up-to-date information and can offer professional advice and guidance.
For credentials: my wife applied for (and obtained) the 'family of a settled person' visa last year, and we engaged a barrister to help us prepare the application and prove that we met the requirements.