The word "eidetiker" has German origins
It is from the German term Eidetiker. (A distinct feminine form exists in German, Eidetikerin, but in English "eidetiker" seems to be used for men and women alike.) The suffix here is not -ker, nor -tiker, but -ik-er: the suffix -ik (related to English -ic and -ics, ultimately from Greek) followed by the suffix -er (related to English -er).
The latter suffix here functions to derive a (masculine) noun referring to a person from the abstract noun Eidetik (in German, words ending in -ik are generally nouns, and correspond to adjectives ending in -isch). The use of the suffix -er in German is explained some by the Dartmouth German Studies Department website a little ways down the page. Apparently,
While this category of "-er" (or "-erin") nouns ordinarily derives
from a verb, there are exceptions...
I guess this is one of the exceptions. Other similarly-formed words in German are Chemiker (chemist) and Physiker (physicist), but these aren't used in English. This suffix or group of suffixes seems fairly productive in that language: Googling supplies a fair amount of examples of Psychokinetiker being used in German text. You are the only person I've found who has used this last word in English text.
Anyway, I know barely any more German than you do, so the above is just what I've gathered through web search and looking at free online German dictionaries. If you want really reliable information about the German word, the place to ask is the German Stack Exchange.
The use of "eidetiker" in English
Wordnik gives several examples of the word "eidetiker" being used in an English context, and more can be found on Google. For example, the term is used in this article on the Stanford Encyclopedia of Philosophy: Other Quasi-Perceptual Phenomena (Supplement to Mental Imagery). Whether or not it remains "purely German" in such instances is a matter of opinion. I'll note that the orthographic treatment of the word in these examples is not purely German; in German, nouns are always capitalized as a rule. There are many psychology terms that originated in German, but are now more or less naturalized in English.
I have found references online to several specialist dictionaries (of English) that list the word: the Oxford Index gives the following definitions:
A person who perceives eidetic images.
–A Dictionary of Psychology, third edition, by Andrew M. Colman
Person with the power of forming eidetic images.
–The Oxford Dictionary of Philosophy (2 rev. ed.), by Simon Blackburn
I found one site that lists as an English translation the (supposed) word "eidetician." But in fact, "eidetician" seems to be barely used at all; in fact, I can't find any examples of it being used with this meaning (it seems to be used sometimes to mean "one who studies ideas," as in this article on the Internet Encyclopedia of Philosophy, or this article that discusses Socrates).
Other "-iker" words used in English
Using findwords.info, I found the following terms:
I don't think "-iker" is a productive ending in English
On the one hand, there seem to be plenty of opportunities for parallel formation, so you might consider it potentially productive because of that. On the other hand, until today I wasn't even aware of this ending for nouns, so if you form new words with it you'll probably encounter more than a few confused people who aren't aware of what you're modeling your coined words after.
I think the following categories of nouns are more productive, although you might find that your formation of psychokinetiker works better to convey your intended connotation than any of the more productively constructed forms.
English has -ician and -icist nouns
"Psychokinetician" and "psychokineticist" are other possible ways to form a noun from "psychokinesis" that I think have more precedent. However, I'm not sure how well either of those formations would convey the idea of "a person with the supernatural ability of psychokinesis": -ician and -icist nouns often describe professions, although some uses of magician perhaps can be interpreted as referring to an ability.
English also has "-icker" nouns, spelled with "ck"
Since -ic and -er both exist as suffixes in English, there are words with this combination of suffixes that aren't derived from German. However, words formed within English from -ic and -er are typically spelled with -icker: a k is added, but the c is also kept. (The current pattern for using -ic and -ick- spellings in English became standardized relatively recently: in older stages of English, it was common to use the spelling -ick word-finally, and the Oxford English Dictionary also shows some historical c-less spellings such as musikers and musyker.)
The same spelling alternation shows up with some words that don't contain the suffix -ic, such as picnicker from picnic and frolicker from frolic.