The page linked above suggests using this repeater design:
X = wall, d = Door, N = plate tied to north door, S = plate tied to south door.
XdX
XSX
XNX
XdX
I think this repeater is a little too fast for the counters to handle; the next repeater tick often comes in before the animal in the counter advances onto the next hatch, making it skip ticks. This would be fine if it skipped a fairly consistent average number of ticks long-term, but, while I haven't tested this, it looks like the number of ticks skipped is pretty random, depending on the mood the pathfinder is in. Also, the repeater cited might not even work if the animal skedaddles out the door fast enough. I lengthened the hall to slow down the count and added pet-blocked doors, as follows:
D1 D2 P4 + + + + + + + + + P1 D3 D4
Ds are doors. D2 and D3 are pet-impassable (but not forbidden; this takes advantage of a pathfinder quick). P1 and P4 are plates triggering their corresponding doors. The ends of the hall are both open to the rest of the fort so the animal will run back and forth as the outer doors actuate. You want to make sure the animal has somewhere it wishes to be; I'm not sure whether a meeting hall is good enough or whether a master/trainer/parent works better. Also, if the repeater gets too long, the animal won't hit both ends.
Some math:
I think that chaining either a 4- or 5-hatch version of the counter is the most efficient way to make your clock last as long as possible. This is based on the following equation:
(clock overflow ticks) = n ^ (c/(n+2)) = e ^ ( (ln(n)/(n+2)) * c )
n is the number of hatches in a single counter, and c is the number of total levels you have to build from Bidok's designs, which I figured was a decent measure of the overall effort you have to put into building the thing. If you want to maximize your clock's tick count for minimal effort, you wish to maximize the quantity (ln(n)/(n+2)). The best value is n=4, with n=5 not far behind. This may interfere with efforts to arrange your counter sizes to come up with a clock period close to a year, though.
Summary of above: 4 hatches per counter seems to be the way to make your clock period last longest with the least effort.
Even with the slow repeater, the first counter in the chain still seems to (mostly consistently) miss a tick when the animal is on the return leg to the top of the counter. You might get around this by simply making the first counter one hatch smaller than your design calls for.
To get the right period for your clock, you may have to do some interesting factoring instead of sticking solely to the 4-counters, as they will probably require extra logic to "decode" the count to find out when it hits the correct point. I think it is a better idea to deviate somewhat from the 4-counters-only design and make it so your final counter in the chain emits a single output tick at the one-year mark. Build about 3 of the 4-counters in a chain, figure out how many days it takes the chain to overflow, and then choose C1, C2, C3, etc. for the hatch numbers of the counters in the chain so that:
360 days ~= C1 * C2 * ... * Cx * (overflow time in days)/(4*4*4)