The OP never suggested changing his logbook. I don't see a huge problem with putting "more than 1000 hours" or "1000+ hours" on a resume, even if the total is several thousand. It's the truth, even if slightly evasive. I'm not sure I agree with his premise that employers are looking for less time, but that's up to him. If asked during an interview for his exact total time, he has to come clean and give a more or less exact number, though.
In the US, logging time in a logbook is not mandatory, and the only time that is required to be logged is that which will be used for a rating or certificate. I haven't bothered with a logbook for many years, and if asked for my exact time I couldn't say with certainty. Once it gets over 10,000 hours it really doesn't matter much what the exact number is, and most employers here know that. In general the only people who put exact hours on a resume are those with only a thousand or so, and I think a general round number is sufficient for that purpose. It will soon be obvious whether your time is closer to 500 or 5000, although I've seen a few pilots with documented high time who flew like someone barely out of flight school. How they keep surviving I don't know.