I hope this qualifies:

+$2
Discard any number of cards for +1 Coffers each.
-
When you buy this, you may overpay for it with Coffers. For each Coffers you overpaid, +1 Card at the start of your next turn.
Rules clarification: Strongroom breaks the rule about having to spend your Coffers at the start of your Buy phase. So for example if you have Merchant Guild(s) in play and bought cards before buying Strongroom, any Coffers gained from the previous buys can be used to overpay for Strongroom. However, you cannot spend Coffers for anything other than overpaying (so for instance, if you only have $3 left, you cannot spend a Coffer to be able to afford Strongroom).
The idea here is to be able to convert Coffers into draw using the overpay ability. The card itself is a terminal Silver that can be useful for hitting certain price points early in the game, since you are likely to collide it with your starting Estates. Strongroom is a stop card, so you probably don't want to buy it too often or else you could clog up your engine. Stocking up on Coffers to get the most use out of the overpay ability might be wise (or just spend those Coffers on buying good draw cards). There are good synergies with other cards that give you +Coffers, and especially with Swashbuckler.
This doesn't really need overpay wording, and would be shorter if it just said "when you buy this, return any number of Coffers for +1 card each at the start of your next turn". It also has tracking issues, as there's nothing to remind you of the card draw, so it should either draw at the end of the turn you bought it, or gain Horses.
Finally, the on-play ability is too strong, being able to gain 4 Coffers per play. Imagine a Strongroom-only play:
T1/T2: Open Strongroom/Silve.
T3/T4: Buy another Strongroom, use Strongroom to gain 4 Coffers.
T5-T7: Use both Strongrooms to now have 12 Coffers.
From here on out, discard 4 if you have Strongroom in hand, otherwise buy Province. You should have 4 Provinces by turn 12-13.
You're right about this not needing the overpay mechanic. As for the other points you've raised, do you think this would sufficiently address them?

I did a simulation with the following strategy:
- buy province if you can (use coffers)
- buy stronghold if you can (don't use coffers)
- buy silver if you can (don't use coffers)
On half of the games, You get 4 provinces by turn 13, and by 15 on 90%
At $5, the average is 15 turns