Could Tamar & Judah Have Been A Marriage?

So I had a question and I was wondering if anyone could answer it for me.

After reading Rabbi Kaplan's Made in Heaven (wrote about this here), I learned that there are three ways in which one can marry a woman:

1) Bi'ah (Sexual Relations)
2) Shtar (Document)
3) For the man to give an object worth the value of at least one prutah to the woman

But there is another idea, apparently rabbinic in nature.

4) To place the two people under one garment (this is like in Megillas Rut, where Rus asks for Boaz to place his garment over her)

If you look at the Tamar & Judah story, it seems awfully like a marriage in those terms.

To begin with, Tamar is veiled, as a bride would be at her wedding:

יד וַתָּסַר בִּגְדֵי אַלְמְנוּתָהּ מֵעָלֶיהָ, וַתְּכַס בַּצָּעִיף וַתִּתְעַלָּף, וַתֵּשֶׁב בְּפֶתַח עֵינַיִם, אֲשֶׁר עַל-דֶּרֶךְ תִּמְנָתָה: כִּי רָאֲתָה, כִּי-גָדַל שֵׁלָה, וְהִוא, לֹא-נִתְּנָה לוֹ לְאִשָּׁה.

14 And she put off from her the garments of her widowhood, and covered herself with her veil, and wrapped herself, and sat in the entrance of Enaim, which is by the way to Timnah; for she saw that Shelah was grown up, and she was not given unto him to wife.

In the verse, it even specifies that the reason Tamar goes out in this way is because she was not given to Shelah as a wife, which suggests she wishes to be someone else's wife.

Obviously the bi'ah portion of this is taken care of, because later we find out Tamar is pregnant by Judah.

Judah gives her his signet ring, wrap and staff. The signet ring and staff would be worth at least a prutah, whereas the wrap is the garment that we see at the wedding (either as a tallis over both bride and groom or the actual chupah itself).

יח וַיֹּאמֶר, מָה הָעֵרָבוֹן אֲשֶׁר אֶתֶּן-לָךְ, וַתֹּאמֶר חֹתָמְךָ וּפְתִילֶךָ, וּמַטְּךָ אֲשֶׁר בְּיָדֶךָ; וַיִּתֶּן-לָהּ וַיָּבֹא אֵלֶיהָ, וַתַּהַר לוֹ.

18 And he said: 'What pledge shall I give thee?' And she said: 'Thy signet and thy cord, and thy staff that is in thy hand.' And he gave them to her, and came in unto her, and she conceived by him.

Obviously this was only in lieu of the kid goat he planned on giving her later, but is it possible that there is a relationship between the narrative and the law? I know Simcha has done research into various relationships between narrative stories and halakha before. Could it be that due to the fact that Judah gave the woman his ring, wrap and staff and slept with her, we begin the tradition of marrying through a ring, saying we must have the two under one garment, and also using bi'ah as an indicator? (Or at least, that these two ideas are linked?) The only part lacking here is the official shtar.

Before you protest that we would not learn anything out of such an untraditional relationship, Rabbi Aryeh Kaplan himself brings down that the source for Sheva Brachos comes from Shimshon's marriage, where he had days of feasting. If Shimshon's marriage to a gentile woman can be considered a forerunner for our own Sheva Brachos, it seems likely that the relationship between Tamar and Judah could possibly be a forerunner to something...