Why would Trump run as a Dem in 2008 when the political winds would likely be blowing in the GOP's direction that year?
In 2008, he'd have to compete against both Lieberman and Hillary if he were to run as a Democrat, and fatigue with the Dems might be significant after 16 years of nonstop Dem rule.
Like I said he probably wouldn't bother to run at all in that year.
In fact I could sketch out a theoretical "Democratic Trump" from this sceario pretty easily:
2008: Romney easily defeats Vice President Lieberman with all the problems that are occurring at this time.
During Romneys first term the Democratic grassroots after ensuing 16 years of moderate Democrats ending in disaster form their own "tea party" reaction. Trump begins to take an interest in politics like IOTL and begins to endorse some kind of crazy anti-Mormon conspiracy theory.
2012: Hillary wins the democratic nomination, however she is forced to move to the left in order to win it, and combined with her atrocious campaigning skills, causes her to narrowly lose to president Romney.
With the Democrtaic establishment being further discredited by Hillarys lost, Trump sees a prime opportunity and decides to launch his 2016 campaign under a populist campaign supporting protectionism and bashing the religious right.