... The 2" .38 +p is more on the .380 side of things, so better to buy an '80s double-stack Beretta.
Yes, no & maybe. It may depend on a couple of things ...
First, the .38SPL may have an advantage when it comes to bullet weight, especially if an expanding JHP is being discussed.
Then, there's also the potential for the frontal shape (meplat) of the non-ex-panded JHP between some .380/.38 bullets, or .380ACP Ball versus a .38SPL WC or SWC.
I once owned a double stack .380 Beretta in the 80's. One of the most annoying guns to shoot because of repeated feeding issues, even when Ball loads were being used.
While the venerable .380 is enjoying a resurgence of popularity because of really diminutive guns, as well as larger double stack guns (attractive to the capacity aficionados), the .380 v. .38 is still an exercise in determining the degree of compromise someone may favor.
I acknowledge I picked up a couple LCP's more than 10 years ago, but that was strictly because I was looking for something that would fit into some short & tight front jeans pockets which wouldn't accommodate my J-frames. It helped that I found the LCP's to be virtually as easy to shoot as my assorted J's, at least on the short range qual courses-of-fire we were running for off-duty weapons, mostly being 3-11 or 3-15yds. Personally, I found the J's to be easier to shoot at further distances, but then I wasn't carrying the LCP's for 'distance' options. One of the compromises.
Yes, I remember listening to Gary Roberts' opinion of .380ACP v. 38SPL as secondary and off-duty weapons many years ago. He also told me how he was a fan of the use of 148gr WC being used by cops in Secondary .38SPL snubs, instead of some of the +P loads. He explained his reasoning as having observed some range testing and evaluations of some cops in a pilot program for potential secondary duty weapons.
The cops participating in the program demonstrated noticeably better controllability and accuracy when using the 148gr WC, versus the +P loads initially used in the testing. That gave them better confidence. A user who has experienced consistency of controllability and accuracy can arguably find it to be an advantage when having to suddenly use it under stress.
Now, while I may favor the nose cavity profiles of some of the more modern .38SPL +P loads, I also like how they've been revised to offer the potential for resisting plugging while still offering the potential for robust expansion. That wasn't exactly something offered by the older JHP and LHP loads, especially in 2" snubs.
Basically, I carry one or another of the modern JHP's in both my .380's & .38's, and have invested the range hours and trigger time with an eye to accuracy of placement. My hope is that the slightly heavier bullet weights of the .38SPL would be an advantage, as might their potential for expansion.
TANSTAAFL, though. In my 37DAO snub I stick with standard pressure 110gr loads, which does put it more in the bullet weight range of the 102 gr .380 loads I often use in my LCP's (when I'm not carrying 85gr or 90gr JHP's).
Controllability & accuracy ... and placement, placement, placement. Regardless of the caliber or capacity. Did I mention the principle of TANSTAAFL?
Just my thoughts, which admittedly have been heavily influenced by my prior LE career, and having trained & served as a firearms instructor for 26+ years of that time. It leaves a mark.
Of course, that doesn't include YOU!!! 
