Can I PCS with a number 3 on my temporary profile?

You won't be able to clear your installation while you're on a temporary profile. If your profile ends before you're trying to leave, you will generally be OK. If not you need to request a deferment.

Unfortunately, I am unable to pull up the regulation that covers the Assignments, Attachments, details and Transfers. However, being an HR Specialist, I am sure you can easily find that title in the Army Publishing Directorate.

That being said, working from memory, there are some profiles and Flags that Soldiers can still PCS with. However, with your having a P3 temporary, you may want to get with your CoC and PCP to find out for certain. I say this because if you were given a P3 Permanent, you would be facing an MEB. With a Temp P3, I am not sure if you could PCS.

** EDIT **
ARMYPUBS finally came back up, but HRC still refusing connections.

SPC (Join to see) - Since ARMYPUBS is back up, I was able to find the info. AR 600-8-101, Out-processing, If a temporary profile is found on a PCS Soldier, it should be determined that the profile is in compliance with the time limitations (including extensions) of AR 40–501. PCS Soldiers with a temporary profile may depart if cleared to do so by the medical facility.

Specific to your question - If you are cleared by the medical side then you can PCS.
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Like MSG (Join to see) I'm having problems accessing those regs (both HRC and Armypubs are giving problems when you try to access them).

I commented on an older post regarding PCSing after pregnancy and vaguely remember reading in the regulation something about a T3 needing a medical review before they PCS.