I strongly recommend against telling them a made-up deadline or talking about the rest of your job search at all unless you have a firm offer in hand from someone else. Employers often ask questions if you bring up your job search and then you have no good options, either lying, walking it back or telling them information they don't need to know.

I'd also avoid asking for a specific date by which you should hear back. If the date passes then any emails you send after that might look pathetic.

It's fine to set a deadline for yourself though. Send your one or two emails and then if the particular date you've set for yourself goes past without a response, count it as over and done with. If they get back to you later, treat it as a pleasant surprise.