I was once involved in a business where I had to write apology/scapegoat/self-extraction letters on an all too frequent basis. There is no real good way to write one, just "less awkward". Invariably, a "it's not you, it's me" methodology can help keep a certain quality of professionalism to it. Distasteful? Perhaps, but one can always wash the bad taste down with a good Scotch or a bottle of fine craft barley pop. Good luck!
Dear Mr. or Mrs. So-and-so;
Regarding our appointment on (time-and-date) to assess or review (blah-blah-blah), I have had an urgent matter arise which is tied to a current contractual obligation that requires my attention, conflicting with our previously scheduled appointment. Regrettably, due to the scope of work involved, I am unable to provide an alternate time and date to reschedule at this time. My sincere apologies.
I cannot ethically ask you to wait for an ambiguous period of time, and I must responsibly ask you to seek another service so that your project is not unduly delayed.
Cordially,
X
It is better to have enough ideas for some of them to be wrong, than to be always right by having no ideas at all.