The kohanim do not have a blanket exemption from shatnez. (Rashi on Arachin 3b makes a point of this.) They are only allowed to wear shatnez in fulfillment of mitzva to wear bigdei kehunah. The mitzva of Yom Kippur is for the clothing of the kohen gadol to be made of pure linen. So the exception to shatnez in other cases is irrelevant.
The reason given for ...
There is an organization in Lakewood New Jersey that gives a week-long training every once in a while. It is called NCSTAR, WWW.SHATNEZ.N3.NET
I know people from other states that arranged to stay in Lakewood for the week and get trained.
Their contact info is:
You can read a short article about them here.
The laws of kilae begadim are taught in the 9nth chapter of masechet kilayim. First the Mishna says that the prohibition is only on wearing as the verse clearly says.
(Kilayim 9.2) הכרים והכסתות אין בהם משום כלאים, ובלבד שלא יהיה בשרו נוגע בהן
Pillows and blankets are included. The Rash explains, according to Gemara Yoma 69a, that from the verses ...
In addition to @user6591's answer, two additional resources that seem worthwhile exploring:
1) TorahLive.com Laws of Shatnez "Learn What and How to Check - Torah Live" offers an online course along with a certificate of completion.
Not sure what the cost is, how rigorous it is, or whether said certificate is widely acknowledged.
2) Yeshivat Pirchei ...
As sabbahillel suggested in comments
there was no concern regarding the prohibition of wearing shaatnez:
Mordechai was permittted to do so since he would be endangering
himself by not obeying the King's request (Sifsei Chachamim to
Megillah 16a, quoted in R Yosef Deutsch's Let my nation live, p.