I'm shooting a single subject in a medium-shot for a series of talking-head videos. The room will be indoors and mostly quiet. I'm thinking of using a wired Countryman B6 and recording to a Zoom H4n. Does anyone think I can get clearer/better quality sound in the same total price range with other gear (another wired lav, audio recorder, or shotgun/other type of mic)?

The B6 is a highly regarded mic, due to it's great sound and extreme diminutive size. There are three sensitivity versions, If you have the standard sensitivity (P4) version, that should be ok. The Zoom H4n has very noisy preamps though and a low sensitivity mic would require more gain and therefore produce a lot of noise.. You may want to think about replacing the H4n with something better, or at least a high-quality front-end pre amp.
Which of the B6 protective frequency response caps you use, depends if it's buried/hidden. But since the B6 is so small, it's practically invisible in plain site.
Many folks use a boom mic as well as a lav (on separate tracks), and choose the better sounding track in post.

In terms of an audio recorder with less noisy pre-amps, any recommendations for something better than the H4n in the same general price range? How about the Tascam DR-40? There's also the H5, but that would be a bit over-budget.

Either a lower-noise recorder or some sort of a low-cost pre-amp (https://www.bhphotovideo.com/c/product/1245016-REG/triton_audio_fethead_phantom_fethead_in_line_preamp_for.html) into the H4n might work.

Most folks that have the Tascam DR-40, seem to like it. Has two mic/line switchable XLR/1/4" combo Inputs, true +4dB balanced line level, and decent sounding on-board mics... As well, safety tracks and single-channel format files are among options. The latest firmware update includes an auto slate tone. Cons: The HP amp is a little anemic and battery life (3-AA) could be better. It can be USB powered though.

Don't forget about the room acoustics. Just because the room is "indoors and mostly quiet" doesn't mean that it has good acoustic environmental properties. You can probably find a tile-wall bathroom or a stairwell that is "indoors and mostly quiet", but that doesn't mean you would want to record there.

Certainly getting the microphone as close to the subject's mouth as possible will mitigate any room acoustical problems. Because you are increasing the "Signal to Noise Ratio" (SNR) of your audio. The closer the mic is to the subject's mouth, the higher the "Signal" part of SNR, so the higher the ratio, which is a good thing. In that respect, a clip-on lav (or even better a "headset" mic) is preferable to a shotgun mic simply because of proximity.

Yes, the room acoustics will be a slight issue as it has a bit of an echo. And a headset mic won't work visually for us, so I do think a lav will be our best bet.

In doing research, I've found some info that the Tascam DR-70D has significantly lower noise, and still roughly within my budget. Has anyone used this?

https://www.bhphotovideo.com/c/product/1086798-REG/tascam_dr_70d_4_channel_audio_recording.html

In terms of an audio recorder with less noisy pre-amps, any recommendations for something better than the H4n in the same general price range? How about the Tascam DR-40? There's also the H5, but that would be a bit over-budget.

In this day and age the original Zoom H4n is a TERRIBLE purchase to make in 2017!

For its time however, when it was first released (we're talking years and years ago, 2009!!) the H4n made sense as people had very few other options if they're looking for an ultra low budget external recorder.

And with the rise of HDSLRs there was a rising rapid need for ultra low priced sound recorders to pair with your HDSLR that was even worse at sound! Thus why the Zoom H4n unfortunately became so very very popular, it just had the right timing when it hit the market.

However the H4n was never intended for this purpose! It is a handheld recorder for the likes of hobby musicians (thus the downright awful pre amps! As drums are much louder than whispered dialogue on a set!). And its ergonomics is awful for a production sound recordist.

In doing research, I've found some info that the Tascam DR-70D has significantly lower noise, and still roughly within my budget. Has anyone used this?

https://www.bhphotovideo.com/c/product/1086798-REG/tascam_dr_70d_4_channel_audio_recording.html

For its price, the Tascam DR70D is the best in its niche for this purpose.

However, I feel the extra money for the a Zoom F4 is extremely worthwhile as it is massively more powerful/featured.

I go into more details covering all the options in my latest blog article:

Which Sound Recorder to buy? A guide to various indie priced sound recorders in 2017 – David Peterson (http://ironfilm.co.nz/which-sound-recorder-to-buy-a-guide-to-various-indie-priced-sound-recorders-in-2017/)

Great, thank you David for that info and the link to the article. I'm definitely not considering the H4n any longer!