If I switch my 39-tooth inner chainring to a 38-tooth will there be a measurable lowering of the gearing? Say I'm in my biggest-tooth/lowest-gear cassette gear and were able to switch between a 39 and 38 tooth chainring, would I feel a difference? I'm exploring chainring options without having to change my bottom bracket parts.

negligible difference.

This is all about gear ratios.

39:27 = 1.444
38:27 = 1.407

which ends up at around 1/4mph difference in the low range.
https://home.earthlink.net/~mike.sherman/shift.html

For any change in front or rear it is simply the difference in teeth divided by the orginal tooth count, expressed as a percent. So 1/39 * 100 = 2.56% difference in gear ratio, speed or distance covered in a crank revolution.

While there's almost no real difference in the gears either will yield (2.5% or so) it can make a difference. Not in the low gear because 2.5% won't make or break anything, but it can affect the spread of the middle gears. Typical steps in a 9 or 10s are between 6 & 10% or so. If you get a gear chart, compare the ratios yielded by both inners, and compare them to the outer's ratios.
One may yield a gear progression about 1/2 step between those with the outer, while the other may be very close to duplicating them. So with one you get a nice batch of very closely spaced mid ratios (those that work well with either chainring) while with the other you don't.

Not a big deal, but if you've ever been on a shallow grade, or working a headwind and wished you had a gear between the 15 an 16, you'll be happy you could find one using the other chainring.

a 135mm bolt circle, 39 is all you have, 130 will let you drop that 1 tooth smaller..

difference is: 39-38 = 1 tooth !!!