Im not entirely sure whether I can say "Bitte Entschuldigung, das war ein Fehler meinerseits" or if there is a better or more correct expression.
Everybody would understand your German sentence, and there’s not much to correct. I‘d just leave out the “Bitte”, it’s redundant or overly polite, and technically, you need a preposition:
Ich bitte um Entschuldigung, das war ein Fehler meinerseits.
While that sentence is fine, it sounds quite formal or stilted. Instead, I’d just say:
Tut mir leid, das war mein Fehler.
Philipp's answer is correct, but your sentence sounds old fashioned/fancy and as you asked for other expressions here are some everyday alternatives
Entschuldigung, das war mein Fehler.
Entschuldigung, das war meine Schuld.
Ich entschuldige mich für meinen/den Fehler.
Das (Der Fehler) tut mir leid.
Bitte entschuldigt meinen Fehler.
or more colloquial
Sorry, der Fehler geht auf meine Kappe.
or even more colloquial
It really depends on the audience and especially on the gravity of the mistake.
The least serious variant, for example if your mistake was to misunderstand the other person, wasting 2 minutes of their time with redundant questions, would be:
Sorry, mein Fehler
That is enough in such cases. If you are talking to older people, or those unlikely to use "modern" casual English phrases, the "Sorry" can be replaced by "Entschuldigung".
If, instead, you didn't respect their right of way and wrecked their car, then the other answers are better :) (but your insurance will not approve, no matter what words you use ;-)