ILR travel to Europe?

Hi, we are thinking of going to Italy for a week in August. Are there any restrictions on US citizen with ILR spouse visa living in the UK to travel to countries with travel corridors such as Italy?
Thanks

Hope it's okay to bump this post.

I am also needing to travel to Italy in August for a family matter. I am looking for advice about whether my US passport effects my ability to travel in the same "free" manner as a British/EU passport holder. I have not travelled outside of Britain since October 2019 and I have a current BRP (FLR).

I have read the guidelines for both the UK and Italy extensively, but I can't find a concrete answer. The gov.uk site says that "Unless otherwise stated, this information is for travellers using a full ‘British Citizen’ passport." https://www.gov.uk/foreign-travel-advice/italy/entry-requirements.

The Italian government website does not mention anything about passport restrictions (only origin and port of call restrictions). As of 1 July 2020 "foreign nationals residing in EU Member States, Schengen Member States, United Kingdom, Andorra, Monaco, San Marino or the Vatican" are permitted to travel to Italy for tourism purposes.
https://www.esteri.it/mae/en/ministero/normativaonline/decreto-iorestoacasa-domande-frequenti/focus-cittadini-italiani-in-rientro-dall-estero-e-cittadini-stranieri-in-italia.html

I'm hesitant to book travel in case my American passport is going to cause me grief at the airports. I know I'm being paranoid, but does anyone know if I am likely to have problems upon arriving in either Italy or back in the UK because I am not a UK passport holder? Or will the scrutiny be solely on my residence and recent travel history?

Any insight or personal experiences would be appreciated! Thank you 

The Italian government website does not mention anything about passport restrictions (only origin and port of call restrictions). As of 1 July 2020 "foreign nationals residing in EU Member States, Schengen Member States, United Kingdom, Andorra, Monaco, San Marino or the Vatican" are permitted to travel to Italy for tourism purposes.

You have your concrete answer right here.

You are a foreign national residing in the UK, therefore you are permitted to travel to Italy without restrictions and without self-isolating.

What matters is what country you are travelling FROM, not what citizenship you hold. If you live in the UK, and have not been in the US in the last 14 days, you will be allowed into Italy without restrictions.

The UK government link states:

The authorities in the country or territory you’re travelling to are responsible for setting and enforcing the rules for entry. If you’re unclear about any aspect of the entry requirements, or you need further reassurance, you’ll need to contact the embassy, high commission or consulate of the country or territory you’re travelling to.

So, it's up to the Italian government to decide whether or not you can enter... and the Italian government says you can.

I am sorry to trouble, @ksand24. I appreciate your response. I think I'm just desperate for affirmation and my uncertainty is creeping in at every step. I am not fond of the idea of travelling, but it's a bit of an emergency trip and I'm under a lot of pressure to get it booked ASAP. Apologies, I'm not thinking straight at the moment. Thank you.