The earth/moon system around the sun certainly changes speed as the earth gets closer to are farther from the sun, but the Earth/Moon orbit the sun together so the direct effect on the Moons orbit regarding the earth's Apogee and Perigee would probobly be small.
I think a larger effect is the tidal effects on the moon's orbit around the earth from the sun.
To look at the math, as simple as possible - when the earth is at it's closest point to the sun, about 91.4 million miles, it's orbital speed is the square root of the change in distance, so it's 1.7% closer, it's speed is about 0.9% faster, and similarly, about 0.9% slower at the furthest point, so, taking 67,000 MPH as the average speed, the earth's fastest orbital speed is about 67,600 MPH and it's slowest about 66,400 - about 1,200 MPH difference, But that effects the earth and moon mostly equally.
If we ignore the Earth's elliptical orbit and set the earth at precisely 93 million miles from the sun, the moon is closest to the sun during the New moon (93 million minus 240,000) and furthest during the full moon, 93 million plus 240,000 miles - so it's a variation of about 0.26% in each direction, 0.52 total percentage change, which is about 0.26% change in speed. 0.26% of 67,000 MPH is about 170 miles per hour, so that is the potential change in the Moon's speed in relation to the earth based entirely on what is essentially the solar tidal effect. The Moon's average orbital speed is 2,300 MPH around the earth, so plus 85 MPH in each direction from the solar tug. - That's enough to throw off the cycle a bit. Not nearly enough to knock it out of orbit (that would be about 41.4% plus on top of the 2,300 MPH to reach escape velocity), but it's enough to create a wobble in the orbit, and my math is probobly pretty rough on this point.
Something else to consider is that Apogee and Perigree operate (almost) on the Sidereal lunar month, or 27.32 days while the full and new moon operate on the Synodic Lunar Month (29.53 days), so the Apogee essentially catches up to the full moon about once a year. I would think that's where the Solar tidal force has a bigger effect, not based on the Earth's apogee, but on the relation between the Moon's apogee to the Moon's position around the earth (full, half, quarter, new, etc). That seems most logical to me anyway, but I'm not 100% sure.
another way to look at this is the Hill Sphere, which is the area of orbital stability around an object. The Earth's Hill Sphere extends about 900,000 million miles. Beyond that, the sun wins. The moon is well within the Earth's Hill Sphere, but not so close to the earth that it's immune to fairly significant wobbles.
anyway - that's my best shot at this question. Orbital Mechanics get pretty complicated with 3 bodies. With 2 bodies, then yes, I think they should be like clockwork.