Another option for the secret bid showcase, if both players tie, award it to the top winner.
I admit I do like the idea of both players doing a separate bid on one of the other prizes as a tie-breaker of sorts.
Another option would be to show a list of 10 possible showcase prices, the top winner would have the option of selecting one first or passing to the other player to select first, and giving the top winner potentially a 1 in 9 chance.
The drawback with a secret bid I realize is it would take away all of the screaming excitement of the audience participation in the live audience in screaming out different bids, but I suppose they could still do so anyway, just without the audience seeing the list of possible prices.
In all cases, the list of 10 possible prices for the showcase, at least one of the prices shown would always be guaranteed to not be over the ARP of the showcase. Conversely, it could be where all 10 are below the ARP as well.