Can I get my promotion effective date pushed back due to error on behalf of unit?

It's not a pain in the ass, it is a 4187 to correct the effective date of rank sent to your BDE S1 and coded in SIDPERS, or IPPS-A, depending on whether NJ upgraded yet.
Once that is coded, your Readiness NCO submits a TL to USPFO-NJ to recover any back-pay due from that correction.
The first step takes about 15 min for your RNCO and abour 2 min for your BDE S1.
The second step takes about 15 min for your RNCO and probably a 1-6 week wait at USPFO depending on their backlog.

FYI, enlisted advancements to E2-E4 are not "automatic" in the Guard, in that they require action by your RNCO and your BDE S1.
Also, that action was delayed due to no one entering your APFT score, which could have been accomplished by your team leader, squad leader, platoon sergeant, unit Training NCO or unit Readiness NCO. And by January or February drill, getting that update should have been your top priority every drill until complete:
Your career and your record are your own responsibility, if it is not accurate, do something about it. And as you move up through team leader and above, you teach that responsibility to your Soldiers. And as a leader, you will ensure, or input yourself, timely and accurate updates of their APFT/ACFT, HT_WT and weapon qual in DTMS.

When I went from E-4 to O-1, I got hosed out of the OE pay scale for two years due to an Admin wonk not doing their paperwork correctly. Took a Board of Correction to fix and a Request Mast to get the Disbursing Officer to cut the check as he decided no junior officer deserved that much money all at once. That stuck with me. As a Skipper, one of my hot buttons was cross threaded records harming my people. My CoC knew that, and when we came across those, the appropriate size bat was used to get things fixed. Run it up the line politely to your SEL to see if that works, BUT you better make sure there is actually a record trail. If no record of the PT, you may be out of luck. That assumes the Readiness NCO isn't gaffing you off to make you go away. BTW, don't assume other SPCs have exactly your circumstance. More often than not, they're pushed back for a reason that isn't advertised, but deserved. Many MILs don't take personal ownership of the glitch when they should. If it doesn't go through, take it as a sad but wiser experience and make sure you look out for Number One. Then when you get to NCO, that same orientation needs to apply in taking care of your folk. Good luck son...

It’s possible, a huge pain in the ass, but possible. Here is the other side of the coin. Do you want to be the pain in the ass!