Can I decline being promoted to Corporal?

If being paid for your responsibility is what you expect, you joined the wrong outfit.
My first military mistake was when I refused to be a recruit Petty Officer in Boot Camp. Paid dearly for that mistake for 2 more months.
The Core Values of the military should make you realize what you do has damn little to do with the paycheck. If this bothers you of course, by all means go to your CSM and tell him you're not interested. You may make it out of the Army before the end of your contract, but most definitely at the end of it. You'll likely not be recommended for re-enlistment.
If you've seen the movie GLORY, you'll remember when Morgan Freeman got his stripes as BN SGT Major. He told his Colonel "I'm not sure I'm wanting this, sir." And his Colonel replied "I know exactly how you feel".
Get used to it, or get out.

Accept the promotion.

I agree with CW2 Carroll, if I was your BN CSM, you wouldn’t have to be asking this question! You don’t have to be THE Leader to Lead. In my opinion, bad choice to even question this. Advancements are almost Alway Earned. This should go into your permanent records so by the the time you are Mature Enough to Take Charge or Command, you’ll be ready. The Military is ALL about Team Work and Consistent Achievements!!! Leading By Example, even at lower ratings!!! Take Charge or get out of the way or get out. Sincerely but Respectfully Submitted! Doc

I posted before finishing, I was about to say out side of the military you are often told/asked to do things you are not hired/paid for. I when I was in I often was in charge without having rank. When that happens it's called being mission oriented. Don't think that what's on your cullor makes you a leader. True leadership comes from the desire to get the job done. If you don't want the promotion now when will you want it? I don't think you should turn it down the chance may not come again.

Isn't a spc and a cpl in the army both E4s and thus paid the same?

The second you do that, you can kiss any chance of getting promoted to Sergeant good-bye. Understand this, although being promoted to Corporal is a lateral promotion, and the pay is the same as a SPC, being a Corporal is being an NCO. Refusing to be promoted to Corporal will tell your SGM you're incapable of handling the responsibility of a Non Commission Officer.

Lead, Follow, or get the hell out of the way!

Take the corporal, or face some serious consequences, with the current drawdown your career is in jeopardy. During my time in service I was acting sgt as a pvt. E-5 in 18 months, in a platoon leadership position for 6months,then PCS Stateside.

YOu should be remove from the promotion list and denied the opportunity to attend any leadership school if you refuse to accept the responsibility of an NCO.

You made this post as “Sgt Josh H”. It seems you want the benefits of being an NCO without taking any of the responsibilities that come with the rank. As far as pay goes, both corporal and specialist are E4, so you are not being asked to do something you are not being paid for.

Smells like bitch in here.