Nope, not even 36mm is possible (on the new market).

I recommend an older Omega AT or Railmaster at 36mm.

Perhaps there exists a Seiko 5 in that size...

 

Nothing wrong, at all, with a 34 to 36 mm Rolex, but if Rolex is not your thing, there are a lot a great "vintage" watches in that size from many brands, such as Omega, IWC, etc.--many have been restored to nearly perfect shape, usually cost less than anything current, and look and wear great--and yes, they still tell time!

 

Tudor still produce some models in smaller sizes. The Tudor Style which may meet most of your requirements (apart from date) is available in 34mm with a variety of bezel options, dial options and colours...

Tudor Style 34mm on Tudors website

Review on Hodinkee

Review on Time & Tide

I have the blue 41mm with fluted bezel, and love it:

 

Tudor still produce some models in smaller sizes. The Tudor Style which may meet most of your requirements (apart from date) is available in 34mm with a variety of bezel options, dial options and colours...

Tudor Style 34mm on Tudors website

Review on Hodinkee

Review on Time & Tide

they still sell the classic Tudor Date Day brand new in certain markets - saw a bunch brand new in Japan and Hong Kong.

the official specs may (?) peg them at 36mm but they actually measure about 34mm in bezel diameter when i had a micrometer handy.

plenty of dial and bezel combinations to choose from. looks great on stock bracelet.

 

Mido Baroncelli 33mm:

Rado Coupole 32mm:

These days you have to look at "Ladies" if you want this size.

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Grand Seiko SBGX263 or older 063 (37mm):

Quartz, but 10s/year and with a nice seconds hand sweep rather than the typical bounce. Gorgeous dial.

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Might I suggest that you will have more luck if you included vintage pieces.

 

Longines La Grande Classique

 

Timex Marlin Reissue.

 

$199 ... hand-wind ... in stock ... and now with a black dial model available ...

https://www.timex.com/marlin-34mm-hand-wound-leather-strap-watch/TW2R47900ZV.html

There's also this classic Weekender ...

There is a 34mm version w/the loudest tick on earth!

 

Max Bill hand wound. Many big name watches from the 1970s and earlier. Solid gold case would be in play but the metal bracelets are hard to find.

 

I agree with all the other suggestions for going vintage. You can get some astounding bargains if you avoid a very few obvious well-known names.

 

Modern mechanical 35mm Constellations can be had for next to nothing used (<$1k from Japan) and should wear similarly to an OP 114200.

 

The Rado Captain Cook reissue is close and one of the most beautiful watches I own

My second one close to that size in my Longines Flagship Heritage, it's also really nice

 

Consider looking for shorter total length watches. It's nor always the case size that determines a good fit. I have a JeanRichard 39mm terrasport that fits my wife perfectly, pretty sure her wrist is under 6". What you really need is a total length of around 44mm.

Prior to finalizing a deal I always ask total length it's what determines a good fit on my 6.75 wrist.

 

Hamilton Jazzmaster Viewmatic 34mm. Marketed as a ladies watch but looks like a classic men's watch to me.

Tissot PR 100 Powermatic 80 Lady 33mm although you might not find a dial/bracelet color combo that works for you.

 

It doesn't have baton markers, but what about the Tudor Black Bay 32 or 36? Blue or black options, two sizes, no date, good quality bracelet.

https://www.tudorwatch.com/watches/new-black-bay-32/m79580-0003

 

Omega DeVille Prestige is available in several smaller sizes if you can deal with roman numerals.

Take a look at a lot of brand's "ladies" ranges. Plenty have factory non-iced models that are more unisex, and you shouldn't need to bother with whatever gender a marketing department decided on for a style anyway.

 

Lots of great buys for 60s or 70s watches that are 34 mm and meet your criteria otherwise, e.eg., Seamaster Deville: