I was looking into the question about Shakespeare. But on the way I found these:
www.ling.upenn.edu/~kroch/hist-pdf/taylor-eng.pdf
english.stackexchange.com/questions/4386/origin-of-different-past-tenses-for-verbs-with-the-same-endings
I'm reminded of two other factors that affect the conjugation of verbs in English: a) newer verbs tend to be weak; and b) when a noun starts being used as a verb, as here, the verb tends to be weak.
So those factors further militate for weak conjugation. The only thing I can find unusual about this verb that might warrant special treatment is that it happens to end in -xt.
As for Shakespeare, he spells the past tense of fix either "fixed" or "fix'd". This is All's Well That Ends Well, Act 1, scene 1:
One that goes with him: I love him for his sake;
And yet I know him a notorious liar,
Think him a great way fool, solely a coward;
Yet these fixed evils sit so fit in him,
That they take place, when virtue's steely bones
Look bleak i' the cold wind: withal, full oft we see
Cold wisdom waiting on superfluous folly.
And this is the same scene:
So show her merit, that did miss her love?
The king's disease—my project may deceive me,
But my intents are fix'd and will not leave me.
These are pentameters (five stresses per line), so we can discern the likely pronunciation from the scansion. Most of the examples I found, irrespective of spelling, don't scan if "fixed" is two syllables. But this one
And our air shakes them passing scornfully:
Big Mars seems bankrupt in their beggar'd host
And faintly through a rusty beaver peeps:
The horsemen sit like fixed candlesticks,
With torch-staves in their hand; and their poor jades
Lob down their heads, dropping the hides and hips,
(Henry V, Act IV, scene 2) only scans if pronounced "fix-ed". So artistic license obviously applies.