Our lovely contributors never recommend hotels that aren't nice or clean -- so no worries there. I'm not sure what you mean by 'freshly updated', e.g within the last couple of months, a year, or ?
Your choice of 4-star hotels in and around the Flea market is restricted. Briefly, there aren't any. The closest thing is the 3-1/2 star Long Beach Marriott.
If you're intent on staying in a 4-star, there is only one in the entire city of Long Beach: the Hyatt the Pike Long Beach. It is, however, not close to the Flea Market which is near the airport -- outside of the downtown area where the Hyatt is.
As far as the bedbug registry is concerned, I wouldn't put any importance in that website for several reasons:
a. Bedbugs don't 'pop up' at hotels due to any cleanliness issues. They are brought in by guests via their luggage. This happens at expensive hotels as well as cheap motels. Once the hotel is made aware of the problem, it is remedied. The fact that a hotel HAD a bedbug report at one time will not be an effective prediction of whether that hotel will have such instances in the future since there's no way to know which guest(s) will be bringing in bedbugs between now and the time you will visit.
b. Some reports on the Bedbug Registry are false, placed by people who have some other negative issue with the hotel. e.g., they didn't like the hotel, the front desk clerk was rude, the wi-fi or parking or restaurant was too expensive, the room didn't look like the one on the hotel's website, etc.,etc
c. And, finally, bedbugs are easily spotted. When you arrive at your hotel -- even if it's NOT on the bedbug registry -- lift the mattress and look along the seams of the underside edge. The bedbugs 'camp out' there and are plainly visible -- looking like little specs.
Have a great time at the Market!