I was frustrated in my recent recruiting. I had targeted a player who I found was considering a Sim team. I reviewed the forums and decided to battle for the player. I understood that Sim teams recruited at the beginning of the signing period, then waited until after signings to recruit again.

I initially got a message making me think I was slightly behind. Additional efforts led to both teams appearing as "being considered." So I spent some more dollars and found now only the Sim was being considered. The recruit had actually refused my home visit after I was "being considered."

Did I drastically misunderstand something? And why was my home visit refused?

Sim does make an additional 'push' after signings, especially on their 'most wanted' recruits.  If the FSS said 'very tight', etc.  the SIm likely spent enough to knock you from being considered. 

If you dont' know where to look for that message, it's at the bottom of the FSS sheet and is updated at 2:30 a.m. and 2:30 p.m. daily

Before battling a sim-considered recruit, I always wait until that first 'Word on the Street' (WOTS) messages comes out.  There are basically 3 levels of these messages, roughly indicating how much effort the sim has dumped into the kid.  If it's a medium or higher level message and I'm at D3, I tread lightly knowing it's gonna cost at least $4K to get ahead.

I'm not sure when you tried to battle the sim, but it's possible considering credit kicked in while you were dumping effort into the kid.  Basically, the sim may have gotten a little "boost" to their initial recruiting efforts because they were on the kid from the onset of recruiting.