The answer given by the user Sarvabhouma although correct doesn't explicitly mentions the taraka mantra exact wording letters.
The original Sanskrit verse (in Padm Puran's Uttar Khand, Ch 243, Verse 35-40) only mention (ते मंत्रं) i.e. Your's (Shri Ram mantr) as follows
मुमूर्षोर्मणिकर्ण्यां तु अर्द्धोदकनिवासिनः ।
अहं दिशामि ते मंत्रं तारकं ब्रह्मदायकम् ॥ ४० ॥
Now what exactly are the syllables of this Ram tarak that Bhagavan Rudr Shankar whispers in the ears of a dying person isn't detailed in the Padma Puran though.
Nevertheless, what exactly does Mahadev Shiv Shankar Bhagavan whispers in the ears of the dying or departed is answered By Mahadev ji himself in the Narad Panchratr, a sacred Vaishnav aagam text.
The Tarak Mantra of Bhagwaan Shri Rama ji that is whispered by Shankar ji in the dying person's ears in the Kashi city is =
ॐ रां रामाय स्वाहा
This is detailed in the Narada Pancharatr aagam, in its Second Night's Seventh Chapters between verse 8 to 13.
श्रीनारद उवाच
प्राणिनां येन मन्त्रेण मुक्तिर्भवति शाश्वती । वाराणस्यां
त्वयोक्तं च तन्मां कथितुमर्हसि ॥८॥अन्यथाऽहं कृपासिन्धो सद्यस्त्यक्ष्ये
कलेवरम् । मां भक्तमनुरक्तं च नाथ मा कुरु वञ्चनाम् ॥९॥
Narada said: You must tell me the mantra that you give to people who
are on the verge of death at Varanasi, enabling them to attain eternal
liberation. O ocean of mercy, if you do not disclose this to me then I
will give up my life right now. My lord, please do not deceive this
soul who is devoted to you.
श्रीमहादेव उवाच
गुप्तं वेदपुराणेषु चेतिहासेषु नारद । पञ्चरात्रेषु
सर्वेषु कथं वक्ष्यामि मां वद ॥१०॥ अहं हत्याभयेनैव वक्ष्यामि गोपनं
परम् । श्रूयतां दक्षकर्णे च न वक्तव्यं कदाचन ॥११॥ मन्त्रोऽयं
मन्त्रसाराद्यः सर्वाद्यवीजमध्यमः । पञ्चवर्गाद्वितीयश्च वर्णश्च
गुरुमान् भवेत् ॥१२॥ पञ्चमे पञ्चमो वर्णो विष्णुमान् ङेन्त एव सः ।
जगत्पूतप्रियान्तश्च मन्त्रः सप्ताक्षरो मुने ॥१३॥
Mahadeva said: O Narada, how can I tell you something that is
confidential to the Vedas, histories, Puranas and pancaratras?
This mantra is the original mantra and it is the essence of all
mantras. It begins with om, followed by the bija, srim, and then the
second letter, ra, that comes after the five sets of consonants,
sounded with a long a, along with the fifth letter of the fifth set of
consonants, ending in the fourth dative case. It is completed with the
word, svaha.
Now there seems to be a difference of opinion between what is "sarvadyabijamadhyamah" = श्रीं or रां.
As per Srila Prabhupad swami of the Gaudiy vaishnav sect it is श्रीं, thus...
The seven-syllable mantra is =
ॐ श्रीं रामाय स्वाहा
(or in its' transliterated form = oṃ śrīṃ rāmāya svāhā.)
ON THE OTHER HAND...
As per Swami Vijnananda of the Ramkrishna Monks Order, its रां, and therefore..
ॐ रां रामाय स्वाहा
(or in its' transliterated form = oṃ rāṃ rāmāya svāhā)
Swami Vijananda ji in his notes explains his reasonings as follows, I'm pasting the screenshot of his translation.

I consulted another translation from a Publishing house in Varanasi called Pracya Prakshans. They also interpret it रां, as follows.

Maybe, Prabupada ji interpreting it as श्रीं, is using some other methodology or sources.
Thus, depending upon interpretation, either the Tarak Mantra of Shri Ram ji as per Narada Pancharatr, is either
ॐ रां रामाय स्वाहा
Or
ॐ श्रीं रामाय स्वाहा
Whatever is the case, Besides Padma Puran, even a Pancharatr aagam called Narada Pancharatr, mention Shiv Shankar ji whispering Shri Rama ji's name into the ears of those who die in Kashi.