According to the Wikipedia article on "O the Deep, Deep Love of Jesus," Samuel Trevor Francis wrote the lyrics to the hymn in 1875, which was subsequently set to the Welsh melody "Ebenezer" by Thomas John Williams. Google Books finds a 1926 edition of Francis's O the Deep, Deep Love of Jesus: And Other Sacred Poems (1926), but doesn't provide a preview or a snippet view of the contents of the volume, so we can't see how Francis himself worded the lyrics.
The earliest version of the hymn that I've been able to find is from Charles Alexander, Victorious Life Hymns (1919), which offers this version of the second verse of the lyrics:
O the deep, deep love of Jesus,
Spread His praise from shore to shore;
How He loveth, ever loveth,
Changeth never, never more;
How He watches o'er His loved ones,
Died to call them all His own;
How for them He intercedeth,
Watcheth o'er them from the throne.
Clearly, the conflict in this verse isn't merely watches versus watcheth three lines later, but watches versus loveth, loveth, changeth, intercedeth, and watcheth in the five lines surrounding it.
The remainder of the song verbeth not in the archaic style. Nevertheless, the first verse includes this specimen of third-person present singular usage:
Underneath me, all around me,
Is the current of Thy love;
and the third verse is rife with relevant instances:
O the deep, deep love of Jesus,
Love of ev'ry love the best:
'Tis an ocean vast of blessing,
'Tis a haven sweet of rest.
O the deep, deep love of Jesus,
'Tis a Heav'n of heav'ns to me;
And it lifts me up to glory,
For it lifts me up to Thee.
From these instances, it appears that Jesus in person loveth, changeth never, intercedeth, and watcheth, whereas the current of Jesus's deep, deep love is (not be) all around, and the deep, deep love itself is an ocean, is a haven, is a heaven, and lifts the singer up to glory (and to Jesus).
On this record, it seems clear that only Jesus in person jusitifieth and deserveth the regal archaiasm of an -eth verb ending; his deep love more pedestrianly merits and receives the plain third-person present form. But that being the case, it is almost impossible to see the phrase "How He watches" amidst the antient dignitie of loveth, changeth, &c., and not conclude that someone—either Francis himself or the publisher of the conjoined music and words—hath blown it. There is not a scintilla of difference in functional meaning between watches and watcheth as used in this hymn.
The more specific circumstances surrounding the occurrences of watches and watcheth here bolster this conclusion. Perhaps most significantly, both verbs are followed by o'er, so it can hardly be argued that watcheth was too difficult to enunciate in the first instance but not in the second.
As scored in Alexander's hymn book, watches sounds as two quarter notes (A–A or C–C or both), while watcheth sounds as either two quarter notes (F–F) or as a quarter note and an eighth-note triplet (D–C-D-E). Either way, I don't see ease of pronunciation as being relevant to the choice of watches in the first instance.
I do see a possibility that because watches appears on a line with no -eth verb close by (changeth is six words away in one direction, and intercedeth 16 words away in the other), a careless author or transcriber might not have noticed the switch in diction at that point. In contrast, it would take a very poor author or transcriber indeed to miss the fact that watcheth appears immediately after intercedeth: even if the sense of the -eth forms were half-foreign to your ears, you could hardly let "He intercedeth, watches o'er them" slide.
A Hathi Trust search yields a copy of S. Trevor Francis, Whence-Whither an Other Poems (1898), which contains the poem "Love of Jesus." This edition came out 23 years after Francis first published the poem but 27 years before he died. Unlike the three-stanza hymn cited above, the poem runs an heroic eight stanzas and attaches two additional verbs directly to Jesus:
Yet He calleth me "His own";
and
When the Royal, Kingly Bridegroom
Hath His stately spotless Bride;
So it begins to look as though the fault is not in Francis's posterity but in himself. In any event, a number of subsequent printings of the lyrics have changed the watches to watcheth, by way of regularizing the diction. In doing so, they have logic and consistency on their side. Those who persist in watches have more than a century of precedent on their side—and not much else.