I think it is important to point out that the Amaunator under the dead deities section here on the EfU website is not the same "Amaunator" as the one worshiped by the Risen Sun Heretics.
The Amaunator worshiped by the heretics is really Lathander, and essentially what is going on is that Lathander is using the heretics to begin a transformation.
At present Lathander is the deity of youth. His outlook on the world is a bit naive and idealistic, believing in looking on the rosy side of things rather than focusing on the bad.
A marked characteristic of Lathander and his church is his constant and endless pursuit of battling evil and the undead. He constantly seeks to make the world a better place, and in his mind many of the evils of the world are laid at the feet of evil deities and their followers.
The transformation from Lathander into "Amaunator" is not so much Lathander actually becoming Amaunator, but rather Lathander assuming a new role and identity within the pantheon. In essence, he's stealing the name of Amaunator for himself.
You might think of it as his way of "growing up." He loses his naive and idealistic nature to be replaced by something more cynical and pessimistic. His rosy outlook is replaced by a more stern and austere personality. His church takes on a more militaristic feel, focusing its efforts less on making the lives of day-to-day people better, and more on the actual eradication and destruction of evil wherever it is found.
In 4th Edition Forgotten Realms Lathander has completed his transformation into Amaunator and is Lawful Good. If you want a general idea of what "Lathunator" looks like then you could draw inspiration from Horus-Re, though dropping the bits about Mulhorand and against ideas of change.
Laws as a whole take on a new importance. Ethical and moral laws in particular, that guard against corruption and sin. Lathunator can have Paladins (because he's Lawful Good), but they are on the smite happy side of things. The law of the land means less than the moral laws of Lathunator. If the government is corrupt, there is no reason the followers of Lathunator should follow those laws and no reason Lathunatorites shouldn't work to overthrow said government. Almost always they'd move to the side of theocracy.
I should note that all of the above is based off my interpretations of Lathander based on the lore and discussions with others. I really love the Risen Sun Heresy, but I am pretty sure you should send in an application.
I should also point out that the server has had those who were members of the Risen Sun Heresy before though the group didn't last long. I remember wern8 playing a Paladin in the group, they also had clerics.
As for the reasons you'd be in the Underdark... I like the idea of coming willingly. Your ideas of the Risen Sun Heresy could be that you are destined to spread the light of Amaunator everywhere, and thus to prove his glory you journey deep into the very heart of darkness itself. After months of travel, losing your supplies, being attacked and fleeing from Drow, having many members with you die, losing your equipment... you and a group of survivors find Sanctuary and then set out to make it your home, believing that "Amaunator" led you to the place and proclaiming it a holy city.
Really, before proceeding, you are going to need to sit down with a DM and talk things out with them. The Heresy might be different on EfU as EfU is not cannon. However, I can say with certainty that in the past Paladins and Clerics of Amaunator have existed on EfU. Your biggest challenge will be discussion with DMs regarding dogma, since you have a heretical cult you obviously do not work from the same dogma as the followers of Lathander.