It is really not that difficult. As long as you have the photo’s signed etc, fill in the forms and a letter (even signed by yourself as MD) explaining you travel a lot (especially if you’re boosting British exports) and need a second passport for when you obtain visas.
They will run their checks and may call you for an interview (unless you apply in person) but it should be no problem at all.
You can prove you travel with visas from your current passport and I can imagine (as can the FO) that if you need to wait 10 days for a visa and need to go to France in between (or even to identify yourself for a bank account) you are blocked.
