I will be answering it based on the Krishna Sandarbha by Jiva Goswami where in he discussed this as part of the actual subject that Krishna is the source of all incarnations which I answered here.
First, he quotes from Brahmanda Purana.
yasmad evam sarvato 'pi tasyotkarshas tasmad evanyatas
tadiya-namadinam api mahimadhikyam iti gati-samanyantaram ca labhyate.
tatra namno yatha brahmanda-purane
trir avritya tu tat-phalam ekavritya tu krishnasy
namaikam tat prayacchati iti vyakti-kriyate cadhikam phalatvam krishna-namna
Lord Krishna is the Supreme Personality of Godhead, and His holy
names, qualities, and pastimes are sublime and unequaled. That no
other holy name of the Lord is equal to the name of Krishna is
confirmed in the following verse from the Shri
Krishnashöottara-shata-nama-stotra, found in the Brahmanda Purana:
"By chanting the holy name of Lord Krishna only once one achieves the same purifying effect obtained by chanting other names of the Supreme Lord three thousand times."
[Next, someone may present the following objection:] Is it not so that
the Brihat-sahasra-nama-stotra (in the 71st Chapter of the
Uttara-khanda of the Padma Purana) Lord Shiva says to Parvati: "O
beautiful goddess, a single utterance of the holy name of Rama is
equal to a thousand utterances of these other names of the Supreme
[To this objection we reply:] This statement of Lord Shiva certainly
explains the superior position of the name of Rama.
But it does not specifically state that the name of Krishna is
secondary. We may also note that in other verses from the Vedic
literature the primary importance of the name of Krishna is revealed.
We may note that because the word "sahasra-namnam" is in the plural, we cannot interpret it to be a dvigu-samasa, and thus mean "of thousands of names of Krishna". For this reason the word "sahasra-namnam" should be interpreted to mean "of those prayers containing a thousand names of the Lord".
He mentions that assuming Rama's name is dropped from sahasra nama stotra because all versions of saharsra namas contains Rama and Krishna names.
Though he doesn't explicitly mention dropping Krishna's name, that is understood since in other places in scriptures, Krishna's name is declared the most important. Else, the statement of Padma Purana and Brahmanda Purana's have no meaning if we keep Rama and Krishna's name inside the Vishnu sahasranamam.
Now the verse about direct comparison is quoted:
padme patala-hande shri-mathura-mahatmye shri-mahadevasyaiva vakye
tarakaj jayate muktih prema-bhaktis tu pavakat iti. purvam atra
mocakatva-prema-datvabhyam taraka-pavaka-samjne rama-krishna-namnor hi
vihite. tatra ca rama-namni mocakatva-shaktir evadhika.
moksha-sukha-tiraskari-premananda-datritva-shaktih samadhiketi bhavah
In the Mathura-mahatmya section of the Patala Khanda of the Padma
Purana, in the description of the holy names of Krishna and Rama, Lord
"From one holy name liberation is obtained, and from the other holy name purification and love of God are obtained."
In other words, by chanting the holy name of Lord Rama, one attains liberation, and from chanting the holy name of Lord Krishna,
one attains pure love of God, which makes the happiness of liberation
seem very insignificant.
Next follows a quote from Vishnu Dharmottara purana:
ittham evoktam vishnu-dharmottarre yac chakti nama yat tasya
tasminn eva ca vastuni sadhakam purusha-vyaghra
saumya-krureshu vastushu. iti.
The potency of the holy name of Shri Krishna is also described in the
"O best of men, on both saints and the cruel hearted(demons) , chanting the holy name of Krishna has equal potency[both attain love of God]"
Next comes a quote from Prabhasa purana(which is a upapurana whose information is given here) which clearly says Krishna's name is the most important.
kim ca, shri-krishna-namno mukhyatvam nigadenaiva shruyate
prabhasa-purane shri-narada-kushadhvaja-samvade shri-bhagavad-uktau
namnam mukhyatamam nama krishnakhyam me parantapa iti.
That Shri Krishna is the most important of all holy names of the Lord
is described by Lord Krishna Himself. As recorded in the conversation
between Narada and Kushadhvaja in the Prabhasa Purana, Lord Krishna
"O Arjuna, of all My holy names, the name Krishna is the most
Gaudiya Vaishnavas think that the original order of the mantra in Kali Santarana Upanishad would be 'Hare Krishna Hare Krishna.....'. The oldest surviving manuscript we have of Kali Santarana Upanisad is a 17th century copy with the commentary by the Advaitin scholar Brahmayogin Ramacandrendra Sarasvati. But, Gaudiya Vaishnavas had been chanting the mantra that begins with Hare Krishna for nearly 2.5 centuries before that. Gaudiya Vaishnav acharyas have given six commentaries on the mantra that begins with Hare Krishna.
By natural assumption that Rama appeared first, while copying or putting in writing form, the order would have been reversed and thus in some manuscripts we find Hare Rama first.
Also, there are cross references for the same mantra in other scriptures, which you can see here.