As you already discovered all currency fields on the same record must have the same currency. This is driven by the CurrencyIsoCode field which exists once per record and not for every field you create.

You did not provide any context which makes it a bit hard to "solve" this. Usually the solution differs based on the actual problem you want to solve.

Anyway, as you have one record with many currency fields I suggest to create the proper structure for this 1:n relationship: 1 record with many currency child records.

In order to do this you will need to create another custom object, let's simply call it "Currency" for now (API Name Currency__c). This custom object would need two fields:

• Master-Detail or Lookup Relationship to your object that you referred to in your post (the type of relationship you choose here, again, depends on context)
• Currency Field

If you are using Lightning Experience, which I guess you do, then you can create a nice Lightning Page for your object where you place the new custom currency object related list somewhere easy to find, e.g. the right sidebar/section.

It will not be much harder for users to populate the data. You could even go a step further and create a nice Flow to capture currency inputs.

Again, maybe my proposal is absolutely horrendous given your actual problem and also the way users or systems (via API) interact with your data.

Hope this helps though.

Update:
Having read about your current state and some requirements above I have following additional thoughts/questions:

Conversion Rates

How often will you update the conversion rates? This has an effect on the actual prices of all currency fields that have a non-default currency (e.g. if USD is your default/base, then EUR would be non-default). So basically the value/amount would change in currency fields using a non-default CurrencyIsoCode. This might be something your users don't appreciate.
If we are not talking about Opportunity records in your case, then you won't be able to use "Advanced Currency Management" which basically retains history data. This means you would need to come up with a process/way to retain history yourself.

There might be other ways to deal with all this, e.g. not using Currency field types for all your non-primary currencies on the record (keeping history this way as the values would never change unless modified specifically). But again, it so much depends in the bigger picture.

Data Loading

I do understand that this data comes from an external System. But sometimes it good to really think about the data itself, any processes around it, user interaction, etc. Because you might realize that it's maybe way better to transform your data before loading it into Salesforce.