Drawing room romance parody in "Hail, Caesar!"

Clarisse Loughrey of The Independent points towards Private Lives (1931) as a possible referent for Merrily We Dance, but notes that it really seems like more of a composite of several different movies from an earlier era:

The reference point for Laurence Laurentz (Ralph Fiennes)'s high-society drama seems a little more obscure; odd, since the other parodies here seem so directly ripped from specific films. Merrily We Dance essentially comes off as an amalgamation of older, refined offerings from the 1930s; a reference to the film adapting a highly successful Broadway production suggests it may be a nod to the work of Noël Coward.

1931 saw the first cinematic adaptations of one of Coward's most well-known plays, Private Lives. ...

Coward originally voiced concerns that [Norma] Shearer would be incapable of handling the play's sophisticated dialogue; to which she replied, "I don't care what he thinks - he thinks in theater terms - I think in film terms. It doesn't seem to occur to Mr. Coward that we both may turn out to be right!"

In turn, Hail, Caesar! sees Laurentz exasperated to discover Western film star Hobie Doyle cast as Merrily We Dance's romantic lead; resulting in the film's standout comedic bit, in which Laurentz tries in vain to teach Doyle the elegance of the Mid-Atlantic accent.

I've come across several other reviewers who have noted that this style of movie seems a bit anachronistic for the time period (e.g., the other quote below.)

As far as who the character of Laurence Laurentz is based on, a lot of people point towards George Cukor or Vincente Minelli:

Hobie’s flamboyant, exasperated director seems partially a nod to MGM mainstay Vincente Minnelli, father of Liza, who directed his then-wife Judy Garland in “Meet Me in St. Louis” (1944). But the ’30s-style black-and-white comedy Laurentz is directing (whose leading lady resembles Deborah Kerr) seems much more like a mashup of Minnelli and Cukor, who directed Kelly’s last MGM musical, “Les Girls’’ (1957).

According to Den of Geek it could be a reference to My Fair Lady (1964). From their website:

Ralph Fiennes has one of the funniest scenes of the movie when he tries to teach Alden Ehrenreich’s cowboy how to properly speak for a drawing room drama. We imagine that a classical Hollywood old schooler like George Cukor had this problem multiple times over the years.

Later:

Helming lavish spectacles of the highest prestige, Cukor had a habit of bringing the best out of Katharine Hepburn throughout her career, as well as knowing how to photograph the most exquisite costumes. Simply look at the below clip from My Fair Lady with Audrey Hepburn (who many at the time thought was miscast as the cockney-turned-posh flower girl), and her in-story troubles with pulling off the proper affectation.

The referenced clip

Although, according to The Verge's review it could be Singin' In The Rain (1952) (emphasis mine):

As much as the Coens reference specific MGM pictures from shot to shot, they're also drawing heavily on Singin' In The Rain, with its sense of looming, sweeping industry change on the horizon. (A lengthy scene where Hobie tries to wrap his natural drawl around his new highfalutin lines is cribbed directly from the film. So is the shot where he drives out of town, neon lights reflected in his car window.)