Eric - You can go to the neighbors pond and catch out 15-25 of his larger sizes of perch as he mentioned. I would also stock some of his fingerlings 2"-4" at about the same time. Only thing is be careful not to get too many introduced right at the beginning. Ask your neighbor how many he thinks you should stock and bring his coments back here and lets discuss them. This method will quicly produce several year classes of perch in your pond. If these adults spawn this spring,, keep and eye on the numbers of small perch hatched, one egg mass from a pair of medium sized perch can easily produce 2K to 4K fry. You should see ths springs hatch at about 2" in August. IN your situation you will definately need a wire mesh fish trap to help sample and remove excess fish/minnows. Go out a buy one or two soon. Bait them with bread.
From what you have said earlier, I understand the perch have not spawned in your location yet so far this spring??. True? Seems to me they should have spawned in May. What are your current surface water temps?
Another thing to consider is to just stock minnow and forage fish this spring. Then this fall and or next spring add perch. This way there will be an abundance of forage fish present and perch growth will be exceptional the first year. These larger perch will be able then to help thin a few of the young of year perch in succeding years. This delayed stocking of perch will also allow the initial stocking of adult minnows to grow to larger sizes and the medium sized perch tend not eat them when smaller minnows are available. You will find that the small and medium sized perch (4" -8") will prefer and best be able to catch the 1" to 2" minnows and many 2.5"-3" minnows will survive to spawn next year if adequate cover is present. In your case, try to get some bluntnose minnows. They grow to 4" long. You might have to trap some from local creeks. Ask the neighbor fish raiser for advice or help with this. Good numbers (80-100)of perch in your small pond will be able to eliminate all but the largest fatheads during fall winter and early spring. As the perch get bigger than 8" then more and more of the biggest fatheads will dissappear. You will find the bluntnose will survive better than fatheads and over time bluntnose will compose a larger percentage of the minnow forage base. Once you introduce bass and they reproduce or become common you will find,,, even with the cover you previously described you expect to have, most all minnows will pretty much be history. Bass will prefer minnows to eating perch and thin them out first. Very difficult in a small pond for minnows to thrive with constant bass AND perch predation pressure. At least that is what we have found out that happens in small NW Ohio ponds like yours.
Ref: your question on stocking time for the smbass. I recommend not putting bass in till small perch are common and numerous ones are showing up in minnow traps and the kids are catching lots of 3"-4" perch. This will depend on how youdo the first perch stocking. When 2" to 4" perch are abundant, it is time for predators or a good thinning of small perch.
I think your kids as they grow older, will soon tire of always catching 3"-5" perch and will be wanting to catch bigger fish. Then you will "mind" too many small fish from slow growth and stunting.