As you probably knows AirPrint doesn't have an option for grayscale or B&W printing.
My printer doesn't have a setting to print in grayscale.
Is there any iOS app that converts a PDF to a grayscale PDF?
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I have not been able to find an app that would do this for you. It is very possible that Readdle's PDF Office can do this for you, however I do not have this app and it is a subscription premium service. I could not confirm from their website or the AppStore for sure that they have the feature to convert to gray-scale, but they bill it as a fully functional and professional PDF suite. Their app Scanner Pro does have the option to save as gray-scale, but the source has to come from your iOS camera. I could not find an option that lets you import a pre-existing PDF into Scanner Pro. Adobe Acrobat isn't even worth mentioning.
I also looked at trying printing directly, but like you said, the printer needs to support conversion to Gray-Scale in order for you to be able to print a color document from the AirPrint as gray-scale. I looked at Readdle Printer Pro, however that did not give the option to convert to gray-scale either.
That being said, if you have a Mac as well as an iOS device, I can provide you with a bit of a convoluted work around.
If the PDF is on your iOS device, you will need to transfer it over to your Mac. You can accomplish this using iTunes, or emailing or messaging the file to yourself, etc. Once the document is on your Mac:
The PDF will be converted nicely to a gray-scale document which you can then print.
Sorry I couldn't come up with an actual iOS app that was capable of this.
I have no association with Readdle other than I have purchased several of their apps for use on my iOS devices, which is why I mentioned them in the post. They build very good apps with functional UI's that are pleasant to work with, which is why I chose to buy these premium apps instead of using free alternatives.