Such an individual could be described as a cynic, or a cynical and captious person.
Caveat: when expressing such an evaluation to royalty, care must be taken, as unforeseen and quite gruesome consequences can result. First, locate your points of egress. Second, ensure no sharp or blunt objects are within the royal reach, e.g., daggers, scepters, falchions, maces, guillotines, etc. Third, the evaluation should be accompanied with much bowing and scraping.
e.g., "My Queen, I find your rejection of my suggestions for a wardrobe makeover utterly cynical and captious! Ah, of course I mean that in the nicest, most flattering way, your Majesty."
cynical adjective
1: captious, peevish
2: having or showing the attitude or temper of a cynic
cynic noun
2: a faultfinding captious critic (especially: one who believes that human conduct is motivated wholly by self-interest)
captious adjective
1: marked by an often ill-natured inclination to stress faults and raise objections
2: calculated to confuse, entrap, or entangle in argument
(all linked definitions courtesy of Merriam-Webster online)
Best Answer
Andrew suggested a good word choice to use in an alternative to your example sentence, which would be something like this:
Can the problem be circumvented?
or
Is there a way to circumvent the problem?
As for a one word substitute for "any way around it", you might consider alternative.
Your sentence becomes Is there an alternative to this problem?
I would suggest a slight modification of this, though, which might not exactly answer your question:
Is there an alternative to addressing/facing this problem?