There is a famous story that when Lord Rama met with Sabari she offered Him some fruits but in order to know if they are sweet she bit them before giving to Him. This website says that the story is in Padma Purana: http://www.valmikiramayan.net/aranya/sarga74/aranya_74_frame.htm However, I could not find anything to this effect in either in Padma Purana or in Valmiki Ramayana. Anyone knows the source?
As you mentioned in your post, there is no reference to the incidence of Lord Rama met with Sabari and she offering him tasted berries, either in the Sanskrit Valmiki Ramayana or Awadhi Tulsidas Ramacharitmanas.
However, there is one more version. Odia Ramayana is written by Balaram Das.
Around the same time as Tulsidas wrote his Ramayana, Balaram Das wrote his Odia Dandi Ramayana.
Here we find the story of how Rama encounters Shabari and she scours the forest to find sweet mango (think of it as berries) for him.
He eat those with "teeth marks" indicating they have been tasted and rejects those without teeth marks. This is probably the first time we get the story of "tasted" or "saliva-soiled" fruits being served to Rama that he accepts.
Of course, the rational behind this can be as as follows:
In traditional societies, food that is tasted is soiled and so cannot be eaten by members of the upper caste. By eating soiled food, Rama indicates how he does not care for caste hierarchy.
If someone can read Odia language, here is link for Odia Ramayana.
From the same article by Devdutt Pattanaik:
Translation Shabari Episode of 15th century Odia Dandi Ramayana of Balaram Das by Prateek Pattanaik (@shrijagannatha):
The Sabarani was a person without knowledge,
Eating a portion, did she know the good and bad.
She gathered the tasty fruits thus,
Leaving apart a Sundari mango, without biting into it.
Within her lotus heart, she thought,
My Lord will eat this fruit,
And this thought did fill her heart.
The supreme one did eat them,
Those mangoes that had an imprint of her teeth.
He returned the Sundari mango to her,
Finding it bereft of any marks.
Sabarani joined her palms,
"I did save one for you,
Why else would I leave a good one apart?"
Said Rama, hearing her answer,
"How did you come to know of its taste?
If it is good indeed, why did you not taste the fruit?
Do not serve me untasted food.
For if I eat this, will I not contradict myself?
In all those mangoes with marks of your teeth,
I found that they tasted great.
I shall not consume this unmarked food."
Saying so, he stood up from his seat.
Lakshmana alone ate the ripe food.
While writing another answer, I searched if this story (Shabari feeding Rama 'tasted' berries) is mentioned in Sri Ranganatha Ramayanam, but it's not there. It just says that Shabari offered various fruits to Rama and then he just ate them:
By the way, where is the scene of Rama eating fruits and eatables bitten and tasted by Shabari?
Has this old woman not offered the fruits to the guests, or just said that they are available?
We listen many stories and see many pictures telling that she gave many fruits after biting them with her own teeth to test the taste.
All that is not in Valmiki Ramayana.
This amplification is from Padma Purana.
phalaani ca supakvaani muulaani madhuraaNi ca | PS
svayam aasaadya maadhuryam pariikShya paribhakShya ca |
pashcaat nivedayaamaasa raaghaabhyaam dhR^iDhvrataa |
PS: I'll update my answer when I locate those exact verses or their translation from Padma Purana.