Is it proper to use the word "bandwidth" as it relates to time allotment?

Bandwidth, literally, is the amount of data that can be passed along a communications channel in a given period of time. On the other hand, in business jargon, it is sometimes used to refer to the resources needed to complete a task or project. Clear language is important for clear communication. So it may be better to avoid jargon if you are not sure they will understand you.

I happen to find this use of the word irritating, but there is no doubt that it is widely used, at least in technology companies.

As you indicate, it will be misunderstood in some quarters, which might be a good reason for avoiding it.

The question of whether or not it is "proper" is meaningless, unless you define the particular arbiter of manners who you want to defer to. There is no authority for the English language.

Lacking bandwidth is generally interpreted as lacking resources. In your case, you mean time, but this is not a given, so it is likely to cause confusion.

Also, "lacking bandwidth" is a term the programmers in my group use to derisively describe folks with diminished processing capabilities.

I would have interpreted this statement to mean that you were too technologically (or cognitively) primitive to be able to complete my request, not that you simply couldn't schedule it.

"Bandwidth" is an expression inferring available "processing power" or "throughput"; how much you can do in a given time. Thus, it is perfectly acceptable in my mind to use the term figuratively to refer to your business' ability to take on more work, roughly synonymous to "manpower" or "man-hours".