Could the Confederates have done better at Gettysburg if they had put cavalry in front of, rather than behind, Pickett's charge?

Lee didn't have a large body of cavalry available for such an attack, because Stuart's Cavalry was trying to get into the Union rear and disrupt it at roughly the time that Pickett was charging up that hill. That was a better use of cavalry in the first place, given the dug in position of the forces on top of Cemetery Ridge, but from Lee's perspective it was an unfortunate failure.

From the Union perspective, it was a successful engagement. It kept the cavalry from disrupting the forces engaged with Picket's division.

The East Cavalry Field fighting was an attempt by Maj. Gen. J.E.B. Stuart's Confederate cavalry to get into the Federal rear and exploit any success that Pickett's Charge may have generated. Union cavalry under Brig. Gens. David McM. Gregg and George Armstrong Custer repulsed the Confederate advances.

No they would not. Cavalry in the Civil War was very vulnerable to fire and was never able to attack formed infantry with success until armed with repeating rifles late in the war. Even then most of these attacks were performed dismounted, with only a small strike team charging mounted.

There was actually an incident in the battle that shows what a debacle this could be. Elon Farnsworth was ordered to attack a skirmish line of Confederate Troops: (from WIKI)

After the collapse of Pickett's Charge and the defeat of Maj. Gen. J.E.B. Stuart's Confederate cavalry on July 3, the third day of the Battle of Gettysburg, Brig. Gen. Hugh Judson Kilpatrick, commanding the 3rd Division, ordered Farnsworth to make a charge with his brigade against Confederate positions south of the Devil's Den area of the battlefield, below Little Round Top. Farnsworth initially balked, arguing there was no hope of success, and only agreed to it when Kilpatrick allegedly accused him of cowardice. Farnsworth made the charge, against elements of John B. Hood's division, under Evander M. Law (Hood having been wounded the previous day). Farnsworth rode with the second battalion of the 1st Vermont Cavalry, alongside Maj. William Wells.

The charge was repulsed with heavy losses, and Farnsworth himself was shot five times in the chest. An account by Confederate Colonel William C. Oates claimed that Farnsworth was surrounded by Confederate soldiers and committed suicide to avoid capture, but this has been disputed by other witnesses and discounted by most historians.[2] Kilpatrick received much criticism for ordering the charge, but no official action was taken against him.

The route of the charge crossed the Emmitsburg Road, which was bordered on both sides by a tall wooden rail fence 4-6 foot high. This would have trapped the horses in a killing zone in rifle and cannister range with no way of crossing until men dismounted and destroyed the fence. Pickett's men climbed the two fences. Cavalry could not.

The role of cavalry shifted dramatically after the advent of long range infantry rifles. During the Napoleonic period, it was still somewhat common to have cavalry make charges against massed infantry formations because, at the time, the effective range of an infantry musket was less than a football field, thus the cavalry had very little distance to cover under fire.

By the time the Civil War came around the normal infantry weapon was rifled and used Minié balls which were much more accurate at range. A cavalryman on horseback makes a pretty large target, for this reason cavalry charges into massed infantry almost disappear after this point. Not to mention, after the widespread use of cavalry by Napoleon and the French, infantry weapons and tactics meant to counter cavalry charges had become much more effective. Cavalry continued to serve a number of roles due to their speed, but unless they had the element of surprise they would typically fight dismounted, as infantry.

To answer your question: Cavalry at the front of the Confederate lines, on horseback, would have likely been slaughtered. In fact, cavalry fleeing through the Confederate infantry lines could have hampered their advance and hurt the morale of the Confederate troops, actually making the charge less effective.