Sri saidevo has stated that according to ShrI Abhinava VidyAtIrtha SwamigaL, the term 'prasanna-vadanaM' in the shlokam refers to GaNesha because "it is the opinion of our ancestors that the term 'prasanna' includes a meaning 'elephant' among its many meanings." He has also given the connection between the phrase 'prasanna vadanam' and Ganesha; it is not clear whether this is also from the book or Swamigal's opinion.
The word 'prasanna' in Sanskrit has the meanings, clear, bright, purest, distinct, perspicuous, true, right, plain, correct, just,
placid, tranquil, soothed, pleased, gracious, kind, kindly disposed towards, favourable, gracious, showing favour (as a speech), etc., and in the context of 'prasanna vadana' it will meanpleased, gracious, kind, favourable etc. I don't know if this word can be split as 'pra+sanna', because the word 'sanna'
means, set down, sitting, occupied with, sunk down, depressed, low (in spirits), languid, exhausted, decayed, perished, lost, dead, shrunk, contracted etc. Even if we take the derivation shown in saidevo's post, it omits the important part 'na' and so 'pra'=forward, to extend, 'sad'=droop, sunk and 'vadanam'=face will make up only prasad vadanam. The third method will give the word 'prasana vadanam' and not "prasanna vadanam".
I could not find the words 'sadAdAna', and 'sadAna' in sadAmada and sadAmatta, the word 'sadA" indicates "always" and not to elephant. sadAmada or sadAmatta do not refer to elephants alone, they will apply to any animal in rut. I also could not find the meaning 'caparisoning an elephant' for the word 'sajjana' (actually the word sajjana in Sanskrit will mean good people, as we see in Bhajagovindam 'sajjana sangatirEka').
So, this attempt to make this Sloka appear as one addressed to or referring to the 'elephant-faced' god is rather convoluted and unsatisfactory.
Anyway, for the information of all I will give one or two similar instances where mantras or riks originally addressed/referring to one deity has been highjacked and included, without reason - rhyme is there, of course !, in the mantra for some other deity which was later included in the Hindu pantheon,
1. The mantra, "gaNAnAm tvA..." which is now recited for Ganesha is Rigveda Mandala II.23.1. It was addressed to the Rigvedic deities Brihaspati & Brahmanaspati, but because the word ganapati appears therein, our ancient priests have taken it as good enough for the new deity Vinayaka and gave him another name ganapati as well. (By way of anecdote, I once heard an interesting conversation between two of the vaidikas who form the third or fourth tier in our priesthood - very little knowledge except for saying 'tathAstu', actually such people used to be called 'tathAstu brahmanan' in those days; they were talking in all seriousness, one asking the doubt that he had read somewhere as to how the mantram 'gaNAnAm tvA...' could not apply to Vinayaka. The other one, a little more aged said, "where is the doubt? Don't you see "Ana" in the last line? What other proof is needed to show that it is for Vinayaka? !! 'Ana' in our local Malayalam means yAnai.)
2. The mantra, "kEtum kriNvan....samushadbhirajAyathA:" is the rik at Mandala I.6.3. It is addressed to the deity ushas of the Rigvedic times and the word ketu here means prajna, consciousness, wakefulness; 'akEtave' in that rik means without consciousness, sleepy, and the rik is one of expressing astonishment at all beings waking up from their nearly unconscious sleep state when the deity of ushas rises in the east. But we are happily reciting it thinking that it refers to ketu.
3. In the DurgAsUktam, the following stanzas are riks:
jAtavEdase...
agnE tvam...
viSvAni nO...
These and the stanza "prithanA jitag^m..." form a prayer addressed to agni in the Mahanarayanopanishad. Thus, Upanishads borrow riks and our mantras take from both without looking into the actual meaning.