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Correct, with them not being a public company it's harder to know exactly what the breakdown of their financials are. Them being profitable without expansion costs was reported by Business Insider. But we don't need BI to tell us this, when looking at Klarna's growth in the US and other geographical locations it's been extremely impressive and this comes at a great cost, which would be captured in General Administrative Expenses (which is very high).

You're also correct that it's hard to know how many of the customers are profitable to Klarna, since the pay-anywhere product wouldnt be very lucrative.
This is likely why in their latest capital raising they were valued just over $10bn and not anywhere near APT's valuation.

Nonetheless, I thought it was worth sharing some of Klarna's achievements in support of your earlier comment.