It is not a good idea. Depending on how bad the battery is, the alternator may try to charge this unchargable battery. If the alternator is working at full tilt for very long, it can be damaged. Buy the battery now, even if you must pay a little more and consider it as alternator insurance. Parts and labor for an alternator are much more than the difference in price on a battery.
To answer your question, you might be OK, but the risk of it causing a much more expensive problem is great enough that it would be wise to avoid it.
