Some thoughts ...
Does anyone know if there is a formula for calculating a reduced V1, assuming airfield details (TODA, TORA & ASDA etc) and Vr and V2 are known
Not the way we go about it, so, no. You could figure one out but for no real value and a lot of effort.
Given that V1min is a function of Vmcg, to get a lower minimum V1, you need to lower Vmcg.
No, you can't do that. Vmcg is fixed for the certification conditions.
So it is possible for some performance software to come up with a reduced V1 when performing a reduced thrust takeoff
No, for reduced thrust Vmcg is fixed and based on whatever the rated thrust is. You are confusing reduced thrust with derated takeoff. In the later, the certification basis is varied to reflect a different engine rating. Conceptually, you unbolt a bigger engine and bolt in a replacement smaller engine. To make things worse, you can play with both derated and reduced takeoff together but you start with the derate and then play with the reduced thrust.
BAe146 fire fighting aircraft is taking off heavily loaded with 3000 gallons of retardant on a runway where it would possibly be more prudent to get airborne following an engine failure than to try to stop. To enable this it would be nice to be able to reduce V1 to as close to vmcg whilst still having enough room to get airborne. It is possible there may be software that can give me the information and it would be nice to know if there is and the cost. For a small operator it may be better if there was a formula that could be used to work it out.
Be very aware that a takeoff failure very close to Vmcg is not for the faint-hearted and is guaranteed to get the pulse rate up and away. The story for a given aeroplane is told quite nicely in the AFM. If the present story doesn't suit your needs, you need a smaller engine. Then, again a smaller engine on the quadrapuff probably is a bit of a non-sequiter.
the low V1 won’t be the issue. However, the distance available for 3 engine acceleration from V1 to VR could be limiting on shorter runways.
All depends on the numbers. On some shorter runways, being able to reduce V1 can make the difference if ASDR were previously limiting .....
If you are talking about a balanced T/O V1, it is simply the speed
Although I very nearly did my initial command on the machine, I have never played with the AFM. I presume that the AFM permits unbalancing which would be a necessary consideration to exploit the OP's concerns.
You could lower V1 to Vmcg if you wanted to, but then you'd be going flying in circumstances where it would be more prudent to stop.
While your opinion is respected, I think it, somewhat parochially, throws out the baby with the bathwater.
V1min would favour continuing
Would you put a wager on that ? While acknowledging the OP's jettison observations, there comes into play a consideration of the jettison reliability in extremis.
safety pee
.. as always, wise counsel.