To me it's a reminiscence rather than a quote. And whatever the case it doesn't affect the meaning of John 20:28 at all. Assuming that it does would boil down to committing the same mistake as the WT does when it inserts Jehovah into a OT quote in the NT because Yhwh was in the original(i.e., the text being quoted, not the text quoting); what matters to the meaning of an OT quotation / allusion in the NT is how the NT writer words and uses the quote, NOT what the quote meant in OT context (which the NT writer very often ignores).
Now in that case it is interesting that the LXX of Psalm 35:23 does have the kind of kurios which usually substitutes the Tetragrammaton, only in the first part of the verse (where no divine name is found in Hebrew and inserting one would break the meter):
exegerthèti kurie kai proskhes tè krisei mou ho theos mou kai ho kurios mou eis tèn dkèn mou.
The latter highlighted expression (correctly understood by the LXX as parallel possessives, not possessive + DN) is indeed almost the same as that which is found in John 20:28 (ho kurios mou kai ho theos mou), only the terms kurios and theos come in reverse order.
The insertion of kurie in the first clause ("Rise up, Lord") which seems to substitute a DN although there was none in the 'original' is actually a pretty frequent case (another famous example is in the quotation of the LXX version of Psalm 102 in Hebrew 1). This is just one case among many showing that textual transmission is fluid and diverse, making it impossible to reach 100 % certainty about where the name was or wasn't in "the original", and turning the very concept of one "original" very questionable, and all the more so when important collections rather than individual short units are concerned. For instance, it has been shown that the composition of the book of Psalms includes a DN revision from Yhwh to 'lhym (God) on one important section (42--89) which is presupposed by doublets in the latter section. So the process of writing and revision actually overlap.