You can simply underexpose by a stop and rely on the printer's ability to recover a usable image from the film, of course, but that's going to come at a much higher cost in terms of image quality than requesting push-processing.
One of the neat things about print (negative) film is that it (usually) has a pretty enormous latitude—it can tolerate quite a bit of over- or under-exposure (say, two stops over and one and a third under) before you really can't recover a usable print from standard film processing. With a one-stop underexposure, your picture will be thin (lacking contrast) and grainy. The point of push processing is to overdevelop somewhat to restore the contrast you lost by underexposing.
If you know ahead of time that you will be shooting an ISO 800 film at 1600 (or if the most important images on thee roll were shot at 1600), then requesting a one-stop push will give much higher-quality images. If there are just a handful of images on the roll that are shot at a higher speed and they are no more important than images shot at the film's nominal speed (or your preferred speed for that batch), then the one-stop underexposure and normal processing will just result in somewhat less-than-optimal prints (or scans) for those images.
So if you're using a minilab, you can simply underexpose and let the printer's autocorrection do what it do. Push processing may be an extra-cost expense you're not willing to shell out, and one stop of underexposure isn't earth-shattering with a print film. If the ultimate quality matters, then push processing will result in images with better colour, contrast and grain, but at a cost. It can't be gang-processed with eleventy-seven other rolls unless the processor has scheduled push and pull run times for the machine, so a local lab will have to make time for your job. And even if you are going to be running a full processor load (submitting multiple rolls at once), you're not going to get normal pricing (just in case you might think it works that way all of the time and come in with one roll somewhere down the line). A pro lab probably won't charge quite as much extra for the push or pull, since that's their stock in trade, but then you have to deal with the pro lab base price and turn-around time.
As an aside, non-overridable DX indexing is an unspeakable evil. I rarely shot any film other than Velvia at its nominal speed—it was usually a third under for chromes, a third to two-thirds over for colour negs, and my N was usually a full stop over for the B&W/developer combos I was using. It varied by batch/lot number, but the ISO-rated speed almost never gave me the contrast and saturation I wanted. If the camera uses electrical contacts for the DX code and has a manual ASA/ISO setting for uncoded films, anything from Scotch tape on up will work fine; you only need special stickers if the camera has no way to set film speed manually. I see you've found a source, but needless to say, they're not as easy to find as they once were (but then, neither is 35mm film).