I am new here and I do not have the reputation score that is required to comment. Hence this answer.
As far as I have searched and heard, there is no reference to any scriptures as such. Srirangam as such is extremely revered and is the most important temple for srI vaishNavites. This temple is revered more than even tirumalA by most of our AchAryAs. To this day, this story of how srI ranganAthA came to srIrangam is only told by various srI vaishNavA scholars during their upanyAsams and they do not quote any of our pUrvAcharyAs or scriptures.
The story goes; after brahmA was born, he set out on a search for his creator and couldn't find anything even after several 1000 years of searching. He then went into a deep meditation to understand himself better. He came to learn of srIman nArAyaNA. brahmA expressed a desire to meet him. srIman nArAyaNA manifested in the form of srI ranganAthA in thirupArkadal (ksheerabdi sagar) and appeared in sathya lOkA (brahmA's abode). brahmA did the daily rituals for srI ranganAthA. The story also tells that srI ranganAthA appeared in sathya lOkA along with the praNavAkAra vimAnam that is present even today.
The first king of the solar dynasty, ishvAku had gone to sathya lOkA to pay his respects to brahmA. He expressed interest in performing daily rituals for the beautiful lord. brahmA was fond of ishvAku and asked the lord what to do. srI ranganAtha permitted brahmA to hand over srI ranganAthA along with the praNavAkAra vimAnam to ishvAku. brahmA desired to continue the daily ritual worship for srI ranganAthA and the Lord told him not to worry as srI ranganAthA will return within one day as time works differently in sathya lOkA. By the time this present day gets over for brahmA, 1000 mahA yugAs would be completed in bhU lOkA where ishvAku was taking him; hence brahmA would be able to do is worship the next day as well.
The rest of the story proceeds as you have described in the question with one small difference. vibhIshaNA had to put the idol down in the present day location in order to perform the panguni utsavam (a celebratory festival done for srI ranganAthA in the tamil month of panguni; Feb-March in the English calendar).
When vibhIshaNA was ready to leave after celebrating the festival, a choLA king of the period by the name dharmavarmA requested vibhIshaNA to leave srI ranganAthA here. The Lord also requested the same of vibhIshaNA and consoled him by saying that he would lie in a posture facing lankA. I haven't checked the statistics; but I have heard it is very rare to see any ranganAthA lying facing south.
thoNdaradipodi AzhwAr's pAsuram highlights this
kudathisai mudiyai vaitthu guNathisai paadham neetti vadthisai pinbu kaatti then thisai lankai nOkki kadal niRak kadavuL enthai arvaNait
thuyilumaa kaNdu udal enakku urugumaalO? en seyghEn ulagatthIrE?
The Lord Ranganathan- who has His Thirumudi in the West; Thiruvadi
towards East; back facing North and the Eyes looking to South
(lankA)..(as assured to Sri VisbhIshaNAzhwAn) is of Blue hued Ocean
coloured, Divinely Beautiful Form. Oh! My body is melting.. what can I
do, oh wordly people!
I have heard this legend several times as it comes up in several upanyAsams. It is generally accepted as the truth in srI vaishNaviTe circles.
As said right at the beginning, I only wanted to comment as the answer as such does not add much value to your question.