Dear Kirk (again)
The scenario has you below minima- which you would be if you had the required references and was attempting to land, but then some thing happens that forces you to go around. Be it fire trucks on the runway, another aircraft on the runway (for whatever reason) etc.
The point is you have gone below with intentions to land and now you can not land.
I am not advocating busting minima for fun or not following a missed approach if a missed approach profile can be followed.
In the scenario you have to come up with a good solution fast.
We are now discussing it on an internet forum and can have different solutions and have the luxury of time to discuss them. This way if the scenario should present it self in the future one will hopefully have "experience" in figuring out a solution.
With regards to LNAV- RNAV is not allowed with pilot defined waypoints, but the procedure could have NAV aids and radials to follow. AND this is an Emergency!
On the Airbus you could as an example enter the OEI SID in Secondary flight plan.
On the Boeing you could enter the points after your missed approach (with a discontinuity) and line select/sequence the first waypoint if you go around below minima.
On PROLINE 21 you could do the same as on the Boeing or use FIXes (provided you have enough FIXes available) the latter would require you to fly in HDG but the MAP presentation will be pretty dxxn good.
IF you choose to follow the SID you might not make the required gradient because of you weight or just because you are SE even with the better starting point (V2++) Higher than AFE and at least at the other end of the RWY.
IF you attempt to get back up on the missed approach profile you might not make the required gradient because you are lower and further along i.e passed the MAP.
IF you fly the engine out procedure (EO SID) you already have more altitude and speed than the procedure assumes (as you are already airborne before the runway starts) and this could make up for the fact that you are heavier.
And there are to my knowledge no EO SID with a higher gradient requirement than a SID?
This solution is only do-able provided you have looked at it whist briefing the approach in the first place.