Shares distributed in-kind from retirement accounts are often (perhaps always) recorded the broker as uncovered shares, so there's a good chance that the broker will not report a cost basis for these shares. They might also report these shares as having an unknown holding period.
Ultimately your tax return must report the correct dollar amounts of LT and ST gains. One way (perhaps the only way) to do that would be to enter each share twice as you suggest, LT with the sales proceeds equal to the value on the date of distribution from the 401(k) (the share value reflected on the Form 1099-R) and the actual cost basis (also determined from the Form 1099-R by subtracting the NUA from the share value) and again as ST with the sales proceeds equal to the value on the date of the sale and a cost basis equal to the value on the date of the distribution. It's the method that I thought of independently before seeing that you had suggested the same thing.
Using this method with your example, yes, you would report a LT sale of $100 with $50 of cost basis and a ST sale of $110 with a cost basis of $100, resulting in $50 of LT gain and $10 of ST gain on Schedule D as you indicated. My only concern is whether the reporting on Form 8949 which will show what the IRS expects to see under these circumstances. I would probably include an explanation statement with my tax return describing this sale and why the sale appears twice on From 8949. I have not been to find any guidance from the IRS on how to report this.
