I think in general it is hard for a truly limited bid to be 100% forcing. Another example would be passing a 2♦ Stayman responce with a garbage-type hand.
No the 2nd: It is considered perfectly normal to pass a Precision 2♦ opening, showing typically 4=4=1=4 shape and 10-15, with something like xx xx Qxxxxxx x
Are we aware of the same BBO? The last decade of BBO is a string of bungled chances. If anyone else came along that was even half way competent they would eat BBO's lunch.
The problem is, if you do that this on this hand, you will find partner doesn't beleive you when you hold x AKxx AKQxx Kxx. Don't be the boy who cried wolf. This hand is NOT that good. Not even CLOSE to that good.
The cardinal rule: Do not balance at imps. (Yes, technically this isn't a balancing bid, but it's an analogous situation). They haven't bid 4!S yet. Don't give them a chance to upgrade their stiff !C holding.
As usual, when you are finding yourself stuck for a rebid, the real problem was your previous bid. To join the chorus: X the first time. Having made this bed, you must now lie in it with a pass. Double would show a lot more than this.