Between "lowest" and "refundable"
 

Does anyone know of a way or website that lets you book a fare that is more flexible than "lowest" but does not need to go all the way to "refundable"?

The telephone.

You can force a few fare classes by selecting upgrade options,
W or higher by selecting System region 2 on international trips
or one of the mileage options MH or YB

but the most complete way is what mahasamatman suggested, call

Why are you wanting a specific fare?

Just wondering if I can save some money by giving up refundability but maybe keeping flexibility.

You would need a list of all the fare bases for your city pair, and then go into each one to determine the flexibility (or lack thereof).

what type of flexibility are you looking for?

No change penalties (when origin/destination stay the same). I've purchased such tickets on UA for international travel. Maybe such fares don't exist for domestic tickets?

Other than award tickets, I am not aware of any (for non-refundable fares).
Your first challenge would be to find such a fare rule and a fare class.

I guess to be really specific, the international fares that I was referring to were: date change permitted for free, refunds permitted but with a fee (which is technically different from non-refundable, where you pay the fee but the balance has to be used on another ticket and not recredited to your credit card).

If you want a specific fare class, as I do sometimes for various reasons, you have to call. .bomb won't let you do it like CO and DL do on their websites. I put the request for the ability to book specific fare classes on every post flight survey I complete.

Some B fares are refundable with a penalty. I am not sure about free date changes.

I don't mean this in a mean way, but this seems like the type of thing only WN and F9 type airlines offer.

I've had numerous international B fares that gave that, although I'm not sure if they were corporate-specific fares or generally available fare classes.

In general unless you're getting a very good deal on B fares (ie, good corporate rates) you'd need to be expecting to make a fair number of changes to make up for the difference.

If you're looking to find the fare levels that might meet the conditions you want, FareCompare has a pretty good tool where you can find all the current fares available in the market you are looking at, and will display the rules for each fare level. It's at http://www.farecompare.com/fare-display/index.html, you just type in origin and destination, and can filter by airline once the results come up. THat way, you can see if the fare class with your desireable condition exists, adn then you can give the fare basis code to the agent.

The part of a domestic fare that drives the penalties for change is almost always the letter "N" in the fare basis, meaning, non refundable, and with it, specific penalties (which can vary by market and fare) for changes.

I am not aware of any case where (in the domestic tariff) the N for non refundable, does not drive some penalty for changes on any legacy network carrier in the DFR (domestic fare rule.)

There are cheap refundable fares in many markets, depending on the competition, that are just capacity controlled. The VA, QA...even some with advanced purchase requirements exist. While they maynot have penalties, and are refundable, you still cannot by a 7 day advanced purchase non penalty refundable fare, and then change it to go 1 day after purchase. Rules still apply.