My first reaction is you're trying to micro-track in far too great of detail, although I suspect you were partially 'thinking out loud' in some points of your post. I don't know if there's any off the shelf software that will do everything you want. But with that aside:
we get one big invoice for many unique order ID’s and in some cases we get invoice for “Product Name” X “quantity” without reference to order ID)
If your supplier doesn't provide invoices with something as basic an order ID you couldn't obviously import the data into whatever platform you're using. The UPS fees they're charging you are likely variable so there's no real way you can get your numbers to completely lineup with theirs to do a flawless audit. I know we have a lot of suppliers who struggle to even send us a PDF invoice every month rather than a mailed hard copy and you probably do too. To get the amount of precision you want I don't think is realistic in a lot of the cases.
What I do is simply do a random audit of invoices, and audit a few lines. My experience in many years of doing this is almost all suppliers are honest about doing this and for as many unfavorable mistakes they make, they make just as many favorable mistakes.
payment provider fee
This should be easy enough to add in because everything follows a pretty standard rate schedule. Probably not the most useful thing though because most payment providers tend to have the same rates +-1% so probably consider it by default in any calculations you do.
4)We buy materials to help us fulfill the orders (packages) – this can be one big invoice every 3 months
6)We pay salaries
I don't think you'd want to try and assign these costs on a per product level. Possibly packaging but even that is a slippery slope.
Now if you want a software solution where you can view a group of orders in a certain time frame for a certain supplier, many order management software suites will do this for you (I know ours, Linnworks, will do it) although you're not going to boil down the expenses quite as much as you're looking for.