Scott,
I don't know for certain this is your requirements based on what you provided, but lets suppose you had networks A, B, C behind the SRX that need to reach network Z behind the ASA.
Since you have 3 tunnel interfaces, we'll assume you configured 3 tunnels each with a proxy id of:
st0.1 Local: Network A, Remote: Network Z
st0.2 Local: Network B, Remote: Network Z
st0.3 Local: Network C, Remote: Network Z
For example,
If Network Z routes out st0.1, only network A will work, not B and C.
The workaround here would be to filter base forward traffic from Network B, when Destined to Network Z and route this over st0.2 (and Network C to Network Z over st0.3). This would be a temporary workaround until Junos supports multiple proxy id's on a route based tunnel, allowing you to add, Network B to Z, and Network C to Z on st0.1, then just a destination route over st0.1 would cover all 3.
Again not sure if this is your obstacle, but it may help you regardless.