A – Scott, the first thing to say here is that the answer to your question will almost certainly be determined under UK / Scottish law on the basis that this is probably stated in the confirmation letter. If it is not so stated, UK / Scottish law will likely apply in any event given that the contract is to be performed in Scotland. I advise on matters of Irish law only and therefore I will answer your question as if Irish law applied. Bear in mind, however, that Irish law almost certainly will not apply to this contract and thus you may want to consider getting advice from a Scottish lawyer.
The second point to make is that regardless of the law in this area, unless you have actually entered into the contract with the venue in Edinburgh, you are of course free to walk away from this gig, i.e., if you find the contractual terms unacceptable you could consider not agreeing to do the gig at all or go back to the gig promoter on the point rather than worrying about the enforceability of the clause restricting the band’s activities. Obviously in doing this you will need to look at the confirmation letter to make sure the contract has not already come into existence.
In any event, under the Irish Competition Act 2002 as a general rule all agreements between undertakings which have as their object or effect the prevention, restriction or distortion of competition in trade in any goods or services are prohibited and void. The aim of the legislation is that people should be fee to earn a livelihood and thus unnecessary restraints on trade are ruled unenforceable. There are certain exceptions to this, for example, certain employment contracts and sale agreements however neither of these are likely to be relevant in your particular case.
The key question would be whether it was reasonable of the venue to impose their proposed restraint on trade on the band. If it were the case that the venue planned to spend a great deal of money advertising the gig and thus if there was a competing gig down the road the following week it may result in substantial loss to the venue, then there might be an argument that the clause should be upheld. However, I believe this is most unlikely. Courts generally shy away from any sort of restraint unless they believe it is commercially justified.
Once again, I stress that I have answered this question as if Irish law applied. As UK / Scottish law will likely govern the contract, if you proceed with the contract and wish to do other gigs within the restricted period / geographical area, you should consult a Scottish lawyer.