Linking Syrio to Daenerys' stay in Braavos is a theory; As Syrio had been the First Sword for 9 years, and no longer is the FS in 298 AC, it is possible (if he had traveled to King's Landing shortly after he ended his office as FS, and was subsequently employed by Eddard) that Syrio's employment for the FS began in 289 AC. (298 AC - 9 years = 289 AC). We know that the Sealord serves for life, but we don't know whether a new First Sword is chosen when a new Sealord is chosen, nor does Syrio specify whether the FS who had served before him had served the same Sealord, or the previous one. So these numbers show us that it might be possible that Syrio was the FS from 289 AC to 298 AC, and that he began his employment in 289 AC because the previous Sealord had died.
But not only is this quite speculative, it also tells us nothing about the Targaryens in Braavos and the moment the pact was signed.
From the app, we know that Dany was five years old when Darry fell ill, and quickly "wasted away". That means that Darry likely died in 289 AC (the year Daenerys turned 5), or otherwise in early 290 AC.
This limits the signing of the pact between mid-284 AC (Daenerys' birth) and 290 AC.
Doran's actions tells us more, however.
Arianne's time in the Water Gardens overlapped with the presence of the daughter of the Archon of Tyrosh. She was supposed to go to Tyrosh in the girls place, to secretly meet with Viserys. Thus, the pact was already signed.
“That green-haired girl was the Archon’s daughter. I was to have sent you to Tyrosh in her place. You would have served the Archon as a cupbearer and met with your betrothed in secret, but your mother threatened to harm herself if I stole another of her children, and I... I could not do that to her.”
The children in the Water Gardens are between the ages of five and ten, and so Arianne would have spend her time at the Water Gardens between 281 AC (the year she turned 5, as she was born in 276 AC) and 287 AC (until the moment she turned eleven). Thus, the pact cannot have been signed any later than 287 AC.
That Arianne was supposed to serve as a cupbearer whilst in Tyrosh, is another hint. A search for cupbearers turns up the following ages: 6/7 (Viserys II Targaryen), 8 (Rhaenyra Targaryen), 9 (Elmar Frey), "no more than ten" (Samwell Tarly); 10 (Tywin Lannister; Arya Stark; Aegon III Targaryen), 12 (Daeron "the Daring" Targaryen), 12/13 (Obella Sand). In addition, we have heard of Rhaelle Targaryen (b. 229-233) who became a cupbearer in 239 AC (10 at the oldest, 6 at the youngest). Based on this, it seems that cupbearers are usually not much older than ~10. Daeron the Daring seems to have been a bit old, but he served as a squire as well, and the age of 12 for a squire is very fitting. Aditionally, the text for Rhaenyra states "At eight, like many another highborn girl, the princess was placed into service as a cupbearer …", possibly indicating that Daeron and Obella are likely to be an exception, with their higher age.
So we've narrowed the range to mid-284 AC to 287 AC.
But even if Daeron's and Obella's ages are not an exception, we can still determine the time-range a bit more specific. The Archon's daughter spend time at the Water Gardens. Her time at the Water Gardens overlapped with Nymeria Sand's, who was born in either 274 AC or 275 AC. Assuming Nymeria had indeed been between the age of 5 and 10 when she spend her time at the Water Gardens, that would place her in the Water Gardens between 279/280 AC and 285/286 AC (until she turned eleven).
Which implies that the pact was signed between mid-284 AC and 286 AC.
And of course, this also fits with Quentyn's age, as Quentyn (born in 281 AC), already was at Yronwood when Arianne was supposed to go to Tyrosh. He had been send to Yronwood an unknown amount of time before, at an age described as “young”, “too young”, and “a tender age”, by Arianne, Mellario, and Arys Oakheart respectively. I can imagine that every age is “too young” for a mother, especially since Norvosi usually don't foster their children, and “young” can mean a lot of things. But it is Arys' opinion which is of most interest. Seven or eight seems to be a regular age to make a boy a page elsewhere (although of course, here too are exceptions to be found, like Robert Arryn, who was supposed to become a page at the age of six, though we also know that the circumstances were not ordinary here). Arys, who most likely had been a page somewhere himself, thinks of Quentyn's age at the time as "tender", suggesting that Quentyn had been younger than children usually are when they are sent away to serve as a page. Born in 281 AC, Quentyn would have been 3 to 5 years old in the range we established above (284-286 AC), and based on that, I would guess that Quentyn had not been at Yronwood for long when Arianne was supposed to go to Tyrosh.
(That would mean that Doran send Quentyn his letter concerning inheritance etc. some 4 years after he arrived at Yronwood, as Arianne was 14 years old when she found the letter. Given that Quentyn was so young when he left home, it seems entirely possible (and I'd think even likely) that Doran did not inform Quentyn of the pact when he originally send his son off, as it concerned treason, and Quentyn was being send to a House whose relationship with House Martell was, at the time, still rather troublesome.)