In this article1, Dr. Juris Zarins places the garden at the north-western tip of the Persian gulf, currently submerged beneath its waters.
As seen in the above image, from as far back as 6000 b.c.e the gulfs northern shore began south-east of its current location, with the mouth of a channel that was fed by four different rivers.
Hiddekel and Prat remain the Tigris and the Euphrates. Zarins names Wadi Rimah and Wadi Batin Pishon. It is a fossil river that flowed from the west of the channel, through a region rich in bdellium and gold, (consistent with Gen 2:11-12). It still occasionally flows. He names the Karun River Gihon, identifying the land of Kush with the land of the Kashshites.
During the Flandrian Transgression, the Gulf began to fill with water, reaching its current level around around 4000 b.c.e, which lines up rather well with our dating for the birth of Adam, around 3760 b.c.e2.
What is left then is the understanding of Genesis 2:10,
וְנָהָר יֹצֵא מֵעֵדֶן, לְהַשְׁקוֹת אֶת-הַגָּן; וּמִשָּׁם, יִפָּרֵד, וְהָיָה, לְאַרְבָּעָה רָאשִׁים.
And a river went out of Eden to water the garden; and from thence it was parted, and became four heads.
Which you argue seems to imply that the 4 rivers began at the Eden River.
Biblical scholar Ephraim Speiser claimed that the verse refers to the four rivers upstream of their confluence into the one river watering the garden. One would then read the verse from the perspective of one traveling out of Eden, (which is how the verse reads), rather than from the perspective of one following the river flow.
1. Here is another article that discusses this topic.
2. 2015-5775/6 = -3760/1