If it wasn't for the LED requirement, I would have recommended the MS8233D or MS8233E and various rebadges thereof (ADM01 / ADM02), yes it is "only" a 2000 count meter, but it has 5x20 fuses on both ranges, is physically solid and with rubber boot, APO, backlight, auto ranging, max capture, it's not quite so blatantly "over rated" in terms of it's warning marks etc (specifically, it's CAT II 600V on the volt ranges, and marked as 250V max on the current ranges)
OEM is Mastech in green, custom branded by a few, mainly bside who does it in blue (ADM01/ADM02), and Peakmeter who does it in orange.
Difference between the two models is D/01 has frequency count mode, while E/02 has temperature mode, I think temperature is more useful myself generally.
Price on Aliexpress typically around the $12-14 mark, sometimes you can get them under $10.
Achilles heel for your requirements though, can't light an LED in diode test mode.