You mention CI Studio in your post, but SASServer3 manages web applications in the Finanacial Management solution suite. Customer Intelligence web applications are managed under SASServer6, which I don't see in your list. The two solutions do not depend on each other in any way that I am aware so perhaps SASServer3 failing to start correctly is preventing SASServer6 from starting. However, you don't seem to have a SASServer6 so we need to get to the bottom of what solutions you have deployed on this server. To that end please post back the contents of the status.xml file you have under Lev5/ConfigData on your middle tier. That will give us some idea of exactly what components are deployed on your middle tier.
SASServer12 starting correctly makes sense to me only because I know the SAS web application server startup script has this wierd thing where it starts 1 then 12, 13 14 ... before it moves on to 2, 3, 4. I'm not sure if it still does that in SAS 9.4M4 (started shipping in November), but it did in earlier 9.4 maintenance releases.
There can be many reasons why SASServer3 is not starting correctly so I'm going to give you many things you can investigate.
You inidicate you see this when you use the .bat file to restart the web application servers. Are you certain this is the only time you see this and that the issue does not occur when the server/machine is rebooted?
When this occurs what errors is the SASServer3 log reporting? The log I'm referring to should be under Lev5/Web/WebAppServer/SASServer3_1/logs. SASServer1 could be reporting issues as well.
Financial Management stores some of its content in a database on your compute tier. I don't have the name of that service offhand, but you might want to see if restarting all of your SAS processes produces the same experience or is it only after some time has passed and then you restart only the SAS web application servers?
Is Financial Management still licensed on the system? Could it be that the license for this solution has expired or was terminated, but the solution was not unconfigured? That could cause startup problems for SASServer3 an SASServer4, but I would expect you to see the same problem when the machine reboots and not just when running the script.
As you can see there are a lot of things to investigate. SAS Technical Support is going to be your best option to reach out to for the type of indepth investigation you need. I'm happy to continue providing tips and things I would investigated, but that approach could take us a lot longer than just reaching out to them.