If I had to guess (and I do), then a first cut on something like this is to try and work out what the string is actually requiring. (Better still, would be that you BUY ONE of them and just measure the current. But without that, and nothing more than a sales pitch on Amazon to go on you have to do some guessing. Do you have an ammeter?)
Some "warm white" LEDs, it turns out, are sold for Arduino boards. Ones I see here, LED Sequins - Warm White, use a \$100\Omega\$ series resistor and operate at \$3.3V\$ and use \$5mA\$. From this, I can guess that they actually require about \$2.7V\$ to operate (\$3.3V - 100\Omega\cdot 5mA = 2.7V\$.) That's at \$5mA\$, but I really don't know what they are actually supplying in that string. But I can now guess that they require about \$2.7V@5mA\$.
The article also mentions \$14\$ days of operation at \$25%\$ duty cycle, which works out to \$3.5\$ days. Six AA alkaline batteries, if loaded around a few hundred mA or less, will each provide about \$2\$ amp-hours of life, or about \$10,800\$ Joules. Multiply that by six to get \$64,800\$ Joules total to work with (if our assumption isn't too far awry.) Let's call it \$65,000\$ Joules, in round terms.
From this, I can now estimate that with 60 of these LEDs, the probable current is on the order of \$\frac{65000 J}{60\cdot 2.7V\cdot 3.5\, days} \approx 1.33mA\$. So the voltage is probably a little lower then \$2.7V\$ each, because a lower current implies that much. So let's refine our estimate here, plugging in \$2.6V\$ into this earlier equation to get close to \$1.4mA\$. Call it quits there.
Now we can estimate the current draw from your series-parallel stack of batteries. The current will be \$\frac{65000 J}{4.5V\cdot 3.5\, days} \approx 48mA\$. A quick double-check of the math yields \$4.5 V \cdot 48mA\cdot 3.5\, days \approx 65,300 J\$. So that seems to be about right.
Getting back to the assumption of about \$10,800 J\$ available in each AA battery, I can now see that the loading is indeed reasonably light and that my earlier guess of about \$10,800 J\$ from the datasheet (seen here) might be slightly conservative. It might work out closer to \$12,000 J\$ each. But that only suggests perhaps a slightly higher current allowed in each LED. Not enough to change the basic estimates here more than a little bit.
From all this, I'd guess that you need a power supply that can deliver \$4.5V_{DC}\$ at \$100mA\$ or better, just to be reasonably safe (I've assumed 100% efficiency up to this point and that's not likely. Odds may be closer to 50%, doubling the requirement to 100mA anyway.) More is better. Since this is about half a watt or so, I'd guess most any wall-wart would do, though you should of course read what the rating says. The only problem with those is that their power transformers are often pretty sloppy, with 30% regulation or worse. But it's probably still okay.
So I'd give it a shot, I suppose. I do think you will be just fine. Best would be to measure things, though. (I'm assuming you did work out the battery arrangements correctly.)
