One syllable antonym for "care"?
Solution 1:
"diss" is a truncation of dismiss and means to reject or belittle. Not a perfect antonym but reasonably close.
"hate" is also stronger than merely "not caring", but is one syllable and an opposite of fondness.
"bore" as a verbal form of "boring" is just about right but seems to run in the wrong direction.
"doze" is suggestive of the attitude of someone who doesn't care, and "sorry to doze too much" conveys the idea of inattention fairly well.
"dream" can play similarly to "doze" but with more baggage of a variety of senses of the word.
"scorn" is probably the most direct antonym and is even listed as an antonym of "care" in some references.
Solution 2:
I can't track down a one-syllable word, but there is a (three-syllable) word for this that has been coined (and defined in Urban Dictionary) but not widely used: apathize. Though it's not common, the meaning is easily grasped through analogy with apathy, and since it's intended for use in a poem perhaps it won't matter that it's not a "mainstream" word.
A (rare) example of it in the wild:
On the wall, certificates and plaques offered vain praise. More fitting, Turney thought, was the Weekend Warrior poster that fellow officers had given him on his last birthday. The Warrior cradled a bag of chips and a six-pack, a fishing pole and net, and boasted a cartoon belly that distended an old wrinkled T-shirt. His eyes were apathetic.
Turney related. In this job he tried to sympathize. Tried to care. "The job's a struggle for me," he'd told his minister. "Sometimes it's easier to just apathize."
"Apathize? Is that a word?"
"It is now."
[Source: Dark to Mortal Eyes by Eric Wilson (2004)]
So, within your example sentence:
Sorry to apathize too much.
Solution 3:
Since you are looking for a single syllable word to fit into your poem which you aren't comfortable with sharing, may I suggest the interjection 'meh', which is used to express indifference. Poetically, you could say something like 'Sorry I was meh' even though this is not normally used. It gives the idea that you don't care about something in the sense you described. Alternatively, consider using the verb 'slack', as in 'Sorry I slacked'.
Solution 4:
In the strictest sense I don't think such a word exists. There are merely verbs that are contradictory but not contrary unless they are abstract actions that apply to truth values (agree and disagree). Sit may be contradictory to run but it is not contrary in your sense. There is no dis-run.
There could always be that you've decided to not care and thus override some internal care. In which case ignore, dissociate, disregard, or neglect would be appropriate. The only word I can think of is chill which is of course colloquial.
I'm not too fussed though. Sorry to chill too much
Solution 5:
If you accept a "modern" tone in your poem, would you care for text or internet acronyms/abbreviations? As others, I did not find pure one-syllable words. However, "DC" or "d-c" are standard shorts for "Don't Care" (I could not find any shorter). Two syllables of course, but playing with poetry rules, that could fit. IDC stands for "I Don't Care" (three syllables).
But wait! There is a rare occurrence of a one-syllable version, even stronger: DEC, for "Don't even care" (UD).
How would Nina Simone's My Baby Just Cares For Me sound then?
Baby, my baby DC for shows
And he DEC for clothes
Artistic license DC for standards. At all.