What cartridge, what output?

the area ~ 0.5mV will be fine, there are some out there, Benz, Lyra, Zyx

Technically, you must choose a cartridge that will:
a) not be obviously incompatible with your tonearm, and
b) not be obviously incompatible with your phono stage gain, and
c) not be obviously incompatible with your phono stage input impedance.

Regarding (a)
According to the information posted at http://www.cartridgedb.com/ your Sumiko Premier FT-3 has an effective mass of 14gm. That makes it generally compatible with a wide array of medium compliance cartridges. Anything with a dynamic compliance around 11-15 CU is unlikely to have any obvious problems (which is all those formulae are good for).

Regarding (b)
Your phono stage gain, as the KAB site and Syntax said, should be fine for any cartridge having an output of 0.5mv or above, which means the majority of MC's. Except for the ubiquitous Denon 103 (which might not be a great match for your arm) most really LOMC's cost more than your budget.

Regarding (c)
What input impedance setting(s) is/are available on your phono stage's MC input? This will determine the coil impedance(s) that will provide optimal frequency response in your system.

So, we know you need a medium compliance, mid-output MC with an internal impedance of ____ ohms. There are (probably) hundreds of cartridges meeting these criteria. How to narrow the field to brands and models?

Well, we can't. Without knowing something of your musical and sonic preferences and priorities, recommendations would be meaningless. If you posted this question on some forums you'd receive a slew of answers that all boil down to "I like X, I like Y, I like Z."

Most posters on this forum know better than to offer meaningless answers that can only confuse or mislead. Should I tell you which medium compliance, mid-output MC's I like? Should Syntax? Should Raul or Elinor? None of that would mean anything, because none of us knows what YOU like.
Help us out a bit. Answer the impedance question and talk about who you are, musically and sonically.

Doug's response to the original post is about the clearest, most well-reasoned response that I have read on this subject. Even better, with some modification to component category, it can be applied to other "what do you recommend" type posts. Very well done and excellent advice.

Doug - Thank you very much. I appreciate the information and what you say about preferences.

My BAT preamp manual says the default cartridge loading resistance value for the phono module is fixed at 47k.

I listen to most kinds of music, with some emphasis on classical, but also jazz (especially New Orleans and ragtime and vocal), some country, some pop, some rock (although my rock library does not go beyond the early 70's). I like delicious mid-range, not too bright a sound, deep solid bass. Detail is good, so long as it does not make fun of the rest of my system. My records are mostly in very good condition and I am in the process of buying a VPI cleaner. I have a fairly extensive library of music on my iMac, played through this system and through another one with similar components, but I do like records and I want to find a cartridge that will make the most of the difference between analog and digital.

Dear friend: As I told you the MM alternative is the way to go and in those cartridge different lines ( Clearaudio, Ortofon and Nagaoka. ) you can achieve what you are looking with out any risk passing the a,b,c points that Doug posted.

You say you have a MM cartridge, could you share which one?

There are almost " hundreds " of MM cartridges that are different of what you have and your phono stage seems to me was designed more thinking in MM cartridges than i MC ones due to its 47K input fixed impedance.
You can run too a HOMC but IMHO the MM alternative is the best for you and your budget.

Now ( in your price range and according your phono stage. ), I can't see a MC cartridge that can perform better than a MM cartridge at the same price.

Regards and enjoy the music.
Raul.