Advice on best mid priced MM cartridge choice for Rega P3 (2016)

This may be a little out of your price range, but you might look into the SOUNDSMITH CARMEN MKII.  The OTHELLO is a little less expensive, but I have no experience with that cartridge. You can't go wrong with the CARMEN

I have a Nagaoka and must say they are extremely musical . It's the type of sound that grabs your atention , makes you close your eyes and transport you. 

On detail the 150 is very good 
 It doesn't sacrifice detail at the expense warmness . On musicality the Grados are excellent . From the least expensive Black they sound very good for the price . Problem with Grados is setup . They are a pain to get them tracking correctly and are very sensitive with setup. 

Ortofons tonally are right smack in the middle tonally . On the Bronze and Black if your system is already warm the Bronze will move it to much towards that warmness , so IMHO the Black would be the better alternative . 

The problem with the Black is that its a very neutral and revealing cartridge so you need very good electronics , if not on some recording your system might sound shrill and threadbare . 
I would recommend the Nagaoka . Musicality with a bit of warmness and detail , great tracking and easy setup . Its a bit expensive but IMO worth every penny . Take care and safe bro. God bless
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Check out the Hana line of carts. I’ve been using the eh, yes it’s moving coil, but can be used with a mm phono stage (47khom). It gets hardly anything but 5 star reviews, which tells me it must be pretty versatile. 

Never heard you table/arm, just a thumbs up for Audio Technica Microline’s, I have prior model 440ML. AT’s 540ml price is shown $370. Amazon $250.

https://www.amazon.com/Audio-Technica-VM540ML-MicroLine-Turntable-Cartridge/dp/B01MEI1RPC

People say specs are not a reason to choose, but imaging of a group of musicians, jazz players especially for me, is important because it is so enjoyable, and hearing the skills of individual players reveals their magic. Also appreciating the work of great engineers. The stiffness of the cantilever material is a determinant.

IOW, while checking reviews of sound qualities ... keep an eye on the specs.

Recently, I compared sound and especially imaging from 4 cartridges, using very familiar well engineered evaluation tracks. 3 arm deck allows instant comparisons. Center is phantom created by matching l/r. I can definitely hear the tighter center imaging, and how wider separation ’lays’ the instruments out more distinctly across the stage, that helped by the tighter center as well.

CS 25db; CB 1.5db Shure V15VxMR with new SAS stylus on boron from Jico

CS 30db; CB .75db AT440ML OEM Microline on aluminum

CS 30db; CB .5db AT33PTG/2 Microline on boron (MC, my first, and now favorite).

CS 25db; CS 2db: Shure 97xe elliptical on aluminum

I like the brush feature of the Shures, but both Audio Technicas produced very enjoyable sound quality while producing improved imaging.

I'm suppose wide frequency response, especially lows and their subsequent overtones is involved in the results:

AT33PTG/2 is 15-50,000hz
Shure 97xe is 20-22,000hz 

Specs: directly comparable to each other???, I just refer to what they publish.

I'll second the hana eh recommendation. I use the el version and it is lovely.