It seems unlikely that there's a connection between these two: if modo came from the ablative of modus (which is indeed a common source of adverbs), we would expect modō, not modŏ.
The most recent source I could find on this was the Rev. J. W. Donaldson, "On the Etymology of the Latin particle modo", in the Transactions of the Philological Society from 1854 (!). I'm sure there are more reliable and recent sources out there: in particular, De Vaan might have a much more reliable Proto-Indo-European origin for the adverb.
But Donaldson agrees with me:
With this [evidence] before us, we may as well take leave of the old etymology, as absolutely irreconcileable with the use of the adverb. I will only add, that if modŏ had in the slightest degree the sense of modō, Cicero would not have written (De Orat. ii.34.146.), "qui ea mediocriter modo consideraverit," nor would Sallust have composed such a sentence as the following (Cat. xxxix. 6.) : "Cujusque modi genus hominum, quod modo bello usui foret." And although there is no difference in the forms of the two words, it must not be forgotten, that while the last syllable of the ablative mŏdō is regularly and properly long, the usage of all the poets confirms the doctrine of Probus (p. 1424), and Festus (p. 140, Müller), that the particle mŏdŏ is a pyrrhichius [that is, two short syllables].
And he doubles down on this when responding to critics of his work:
A Member maintained that all the meanings of mŏdŏ could be deduced naturally from the original one of "by measure, only." If this can be done, it is a pity that it has not been done. All the attempts, with which I am acquainted, are utter failures […]
His suggested etymology is from Greek μοὶ δός or the Latin equivalent *mihi dō "give me" > *mīdō > modo, comparing it to cĕdŏ from *ce "here" + dō, and explaining the *ī > ŏ by comparison to *hī diē > hŏdiē "today". But I don't buy his analogy: surely *hī would be in the ablative, *hō, to agree with diē? He has other analogies, but all of them go back to PIE ablaut patterns in verbs.
Anyone with access to more recent scholarship is welcome to supplant this answer! I'm also interested to see if there's a strong PIE etymology for mŏdŏ.