I’ve had 2 UK passports for years for work reasons (visa applications / travel to ‘incompatible countries’ etc). As above you just need a letter from your employer stating the business need.
Last time one of my passports was up for renewal I went to the PP office in Peterborough to get it done same day, and they were happy to do both at the same time (needed forms and photos for each plus the company letter again). Conveniently this means that I now also have consecutive passport numbers and the same expiry dates – which is handy for remembering when filling in forms.
If you do a lot of travel and need a lot of visas then it’s definitely worth doing.
For your visa problem, CIBT are generally fairly capable when it comes to fast turnaround. I think they even publish some specific guidance somewhere for getting your Russia visa application submitted the same day.
Good luck!