I don't have a solution but there is a problem.    

The scenario:  You have a Type A who will re-sign.  You don't think he'll get signed so you cut him loose.  This is because he's a borderline guy and no one wants to give up a pick.  So you wait.   Then, on the last cycle of FA(assuming he's still available), you snatch him up on the cheap.  Now he'll re-sign for another for the same cheap salary.   Maybe multiple years for the cheap salary(I haven't tried). 

Example:
Hardball Dynasty – Fantasy Baseball Sim Games - Player Profile: Howie Etherton

Now Howie isn't the same guy he has been but he's worth 875k.  Because he'll take some ratings hits, I'll run him through waivers after the transaction deadline.  He'll pass through.   Once the season is over, I'll re-evaluate his decline but, in all likelihood, he'll be worth another 875k.   I'll sign him and send him to AAA so I won't be committed to keep him on the BL team(or pass him thru waivers) next season.   Worst case scenario, I've got a cheap injury back-up for next season.  

The only problem I see is that some players are not worth the Type A designation they are given by the engine.

Nothing wrong with what you've outlined.  31 other teams passed on him during FA, so you're really not manipulating anything.  It comes down to supply and demand.  You're the only one who finds him of value.

But he finds him of value because his cost is different.  That's true for all Type A free agents, though.  They are "cheaper" to you than they are to anyone else who would win them.

Exactly.  I wouldn't give up a pick for him at any price either.   But that doesn't apply to the team that cuts him loose.   So the guy will sit there and wait for me to sign him or until the draft prospects show while his ratings decline and he loses the designation. 

Maybe the solution is similar to MLB.   At some point, MLB teams can't sign their FA until May 1st.    Maybe we shouldn't be allowed to sign our Type A at less than their initial asking price before game 20.

If nothing else, don't let him re-sign for the discounted price NEXT season.   I'm not so much bothered by the fact that I was able to get him cheap this season.  As you said, no one wanted to pony up the pick.  But I'm getting him for two seasons at the cheap price(assuming he clears waivers).

But if nobody was willing to give up the comp pick even at $875k this season, it it reasonable to think that anybody would be willing to do it next season when he's a year older and (presumably) further in decline?  Again, I don't see this as a real problem or manipulating anything.  The market has set his price, I'm not sure that it needs to be reset.

I do like the idea of not being able to resign your own departing FA between the start of the open FA period and May 1 (game 27?). 

With one exception: a player who refuses to negotiate an extension the previous season, or who will not negotiate during the exclusive period because he "wants to test the waters.  A team should not be prevented from trying to re-sign a guy who left without giving his original team a shot at him.

He may not be a Type A next season. 

It probably comes back to not receiving a comp pick without offering arbitration.    That just seems like a ***** to program.