Surprisingly, it appears that a counterpart of the Stabat Mater exists, the Stabat Mater Speciosa. It has been found in a fifteenth century handwritten manuscript. This poem describes the feelings of Maria after the birth of her child. It has been set to music by Liszt and Diepenbrock, who both also composed a Stabat Mater Dolorosa.
The version by Liszt consists of no less than 23 stanzas! Three stanzas (8a, 8b and 13b in the following table) have no counterpart in the “Dolorosa”. I think that we may assume that these have been added later by someone other than the original poet of the Speciosa. Diepenbrock also includes the stanzas 8a and 8b, but stanzas  9, 10 and 13 are missing, or left out deliberately by Diepenbrock. Comparing both versions it is interesting to note, that there is a difference in line 1 of stanza 5: “Quis est, qui non gauderet” (Liszt) versus “Quis est homo, quin gauderet” (Diepenbrock) and in lines 2 and 3 of stanza 11: “Pone nostro ducas plagas cordi fixas valide” (Liszt) versus “Prone introducas plagas cordi meo valide” (Diepenbrock).
The “Speciosa” is clearly based on the “Analecta”-version of the “Dolorosa”, and it is easily seen that the rhyming scheme is not followed as strictly as in the “Dolorosa”, especially in the third lines (see stanzas 7 and 8: subditum – diversorio).

The complete text is as follows; the English translation I made is based on a Dutch translation.
I did find a translated rhyming version in German.

Stabat Mater Speciosa
Stabat Mater speciosa Iuxta foenum gaudiosa Dum iacebat parvulus	1	The beautiful Mother stood joyously at the crib in which her child lay
Cuius animam gaudentem Laetabundam et ferventem Pertransivit iubilus	2	Through her exultant soul Dancing with joy Went a song of rejoicing
O quam laeta et beata Fuit illa immaculata Mater unigeniti!	3	O how jubilant and blessed was the immaculate Mother of the Only-begotten
Quae gaudebat et ridebat, Exultabat cum videbat Nati partum incliti	4	O how happy and laughing And exultant did she watch The birth of her divine son
Quis est, qui non gauderet, Christi Matrem si videret In tanto solacio?	5	Who would not rejoice If he saw the Mother of Christ In such comfort?
Quis non posset collaetari, Piam Matrem contemplari Ludentem cum Filio?	6	Who would not jubilant too Watching Christ’s Mother Playing with her son?
Pro peccatis suae gentis Vidit Iesum cum iumentis, Et algori subditum.	7	For the sins of His people Amidst beasts of burden she saw Jesus subjected to the cold.
Vidit suum dulcem natum Vagientum adoratum Vili diversorio	8	She saw her sweet offspring That she adored, crying Swathed in cheap bandages
Nati Christus in praesepe Coeli cives canunt laete Cum immenso gaudio	8a	For just-born Christ in his crib The angels sing joyously And in great rejoicing
Stabat senex cum puella Non cum verbo nec loquela Stupescentes cordibus	8b	The old man stood at his young wife Without speaking, and his heart Filled with unspeakable wonder
Eia Mater, fons amoris Me sentire vim ardoris Fac, ut tecum sentiam	9	Oh Mother, fountain of love Make me feel your ardour Let me share it with you.
Fac, ut ardeat cor meum In amando Christum Deum Ut sibi complaceam	10	Make my heart burn With the love of Christ-God And find grace in his eyes
Sancta Mater, istud agas, Pone  nostro ducas plagas Cordi fixas valide.	11	Blessed Mother, be not harsh Cause your sufferings To be fixed deeply in my heart.
Tui nati coelo lapsi, Iam dignati foeno nasci, Poenas mecum divide.	12	With your child from heaven Let me share my part Of the penance He deigns to bear
Fac me tecum congaudere Iesulino cohaerere Donec ego vixero	13	Make me rejoice with you, and share the adoration of Jesus, as long as I shall live
In me sistat ardor tui Puerino fac me frui Dum sum in exilio	13a	May your ardour fill me May the child be my refuge In my exile
Hunc ardorem fac communem, Ne me facias immunem, Ab hoc desiderio.	14	Familiarize me with this ardour Make that I do not turn From this desire.
Virgo virginum praeclara, Mihi iam non sis amara Fac me parvum rapere	15	Virgin, most exalted among virgins, Be not bitter towards me, Let me take the child in my arms
Fac, ut portem pulchrum fortem Qui nascendo vicit mortem, Volens vitam tradere.	16	Let me have the strength of him, Who by his birth conquers death, And is willing to give his life.
Fac me tecum satiari, Nato tuo inebriari, Stans inter tripudia	17	Let me be with  you fulfilled, Intoxicated with your first-born Under such good omens.
Inflammatus et accensus Obstupescit omnis sensus Tali decommercio.	18	Thus aflame with fire of love, All feelings are silenced By such selflessness
Fac me nato custodiri Verbo Christi praemuniri Conservari gratia	19	May the first-born protect me, And Christ’s word strengthen me, And his blessing save me.
Quando corpus morietur, Fac, ut animae donetur Tui nati visio	20	When my body dies, Then let my soul behold, The sight of your first-born.