Those blood clots are not a good sign. It reduces the odds that this is something the pullet can survive.
At four months, is this her first egg? Or has she laid others? The blood clots indicated injury somewhere in the reproductive canal. Yes, she needs an antibiotic. You can find amoxicillin as "fish mox" online. I suggest you order some ASAP.
She needs calcium to strengthen her contractions to expel all this egg material. It's never easy to pry open a chicken's beak, but that you absolutely must do. Tums will do, but calcium citrate with D3 is more easily absorbed. You can put any size pill into a chicken's moth and they will have no trouble swallowing it.
If you're close to town, hop into
Walmart and grab some calcium citrate from the vitamin aisle. While you're there, slip over to the women's hygeine products and grab a tube of miconazole seven day yeast treatment.
Back to the crisis at hand. Fill an oral syringe with oil, mineral or olive, and insert the tip just inside the cloaca and lubricate the inside so the egg material will slip past more easily.
Provide her with fresh cool water as she should be very thirsty losing fluids due to the blockage. If she gets dehydrated, she won't be able to pass the egg.
Hang onto the miconazole until we see if she can resolve this blockage. It's normal for the crop to back up when there's a blockage downstream. If the crop hasn't gone yeasty yet, you may not need the miconazole, but it will be good to have it on hand anyway.