What to do when pneumatic air cycle kit stops working mid-flight on Boeing 737-800?
Different airlines may have different policies, but I think what my airline uses is probably pretty standard in this regard: no, you can continue at FL 390.

The limitation of not exceeding FL 250 on one pack comes from the MEL, which is a pre-takeoff document.  If the pack is known to be inoperative before takeoff, then the flight has to be planned at an altitude no greater than FL250, which means that the fuel planning has to account for the burn rate at the lower altitude.  However, when a system fails after you are airborne, the controlling document is NOT the MEL, but the QRH (i.e. your non-normal checklist for pack failure).  The QRH from Boeing does NOT direct you to descent to FL 250 for a single failed pack, for exactly the reason you allude to in the question -- you might very possibly have to divert due to the higher-than-planned fuel burn.

So in the case of a pack failure at FL 390, you can continue that flight at FL 390 as planned, but all subsequent flights until the pack is repaired have to be conducted at or below FL 250.

Some airlines may have a different policy, but that's the basic answer from Boeing's documents and philosophy.  Also, fuel permitting, some pilots might descend from FL390 to something a bit lower if they thought that the remaining pack might have an issue, just to provide a bit of cushion in case it does decide to stop working.  Packs fail rarely enough, though, that the risk of the second one failing on the same flight is pretty remote -- which is probably why Boeing is okay with continuing at the high altitude for the one flight, even though if you'll have to fly several more flights before getting repairs, they want the risk minimized for the others.  Risk management.