Is it possible to display the default currency without the currency code while still displaying the currency code for all other currencies?

Yes, you can set this up by updating your "CURRENCY FORMATTING" in your Shopify Admin settings, which can be found in Settings > General > Standards and formats > Currency > Change formatting.

Remove the currency symbol from the "HTML with currency" field and then the currency code of your store base currency will not appear on your website.

In your app's dashboard, select the first option listed under "Display Format", so that all of the other currencies will display their currency code.

We've attached two screenshots for your reference.