Are there any analytics tools to check the best posting time for an account you dont own?
Such as one which analyzes the performance of posts during time certain time periods, and come up with an estimate for which time of day brings the most engagement.
The purpose of this is to analyze the best time to post for an account I do not own, such as for a shoutout.
Only tool I know that does that is ‘Plann’ it’s a mobile scheduler app but can give you the best time to post. Besides that i’m sure you already know to check Instagram Insights.
Would I have to provide account login details to see the analytics of an account?
Yes you do, but its a great alternative if you don’t want to go into Insights
I haven’t the exact answer to your question, but with OwlStat you can analyze a post and its engagement over the time. You don’t have to insert your username or password, just the post URL and It could be useful for this kind of analysis: i.e. you can insert the post you upload on the ‘post analysis’ section, and look at the graphs that shows the likes/comments. So, if you do this with every post you upload you can see the common results on your posts and check what could be the best time to post.
Hope I’ve helped you in some way 
Ps. It’s a function that it’s only in the dashboard. With the free plan you have just 1 analyzable post, but if you want to test this function I can enable you to track more post for testing (free) or I can give you a coupon for this service!