Can I decline being promoted to Corporal?

It's not a promotion, it's a lateral appointment. But you should take it. I was appointed to corporal and it allowed me to learn a lot more responsibility than my peers. Sure, it sucked not getting paid as much as the SSG's that I shared taskings with but trust me, if you are in it for the long haul, it will make you a better leader. You will be a member of the NCO Corps and you will treated as such, so act like it (and not an E-4). Just my 2 cents.

It would be a bad career move if you don't. You know this will go into your record! The military is not about you, it is about the military and what they want. It can be a fiercum beast and not your friend, especially for non-compliants. One bad move may follow you as long as you are in!

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.