Perhaps the Larkin/Lanceryn's bond is with Starsprens and not Madras (Lucksprens)?

[Dofurion]

It's an interesting idea that came to me when reviewing my notes on the Larkins. The body structure of the Madras in the cognitive is very similar to that described for the Starsprens, the latter being much larger and much rarer to see.

[alder24]

42 minutes ago, Dofurion said:

It's an interesting idea that came to me when reviewing my notes on the Larkins. The body structure of the Madras in the cognitive is very similar to that described for the Starsprens, the latter being much larger and much rarer to see.

Chiri-Chiri needs a Mandra, that's confirmed by Nikli. Luckspren was guiding Rysn to Akinah, to the cave under the city and there are hundreds of them around the island. Moreover Luckspren are seen around greatshells, they are bonded with them to help with their mass. Larkin are greatshells too, they need Luckspren to grow to those enormous sizes. But they need a special kind of Luckspren, found only on Akinah. 

Dawnshard ch 9:

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“My father is at Urithiru,” Cord said, gripping her hand again. “Thank you. Yes, that would help. He—” She stopped and looked sharply at the sky.
“Cord?” Rysn asked.
“Spren,” she said. “In the sky.”
“I don’t see any,” Rysn said, frowning and glancing upward. “Did one of the stars move?”
“No, not starspren,” Cord said. “Apaliki’tokoa’a. Lopen called them luckspren.” She frowned. “They are swirling around in the sky, and keep darting toward the ocean, then returning. They dislike that we have delayed. They want us to continue our voyage.”

ch 12:

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“Luckspren,” she said, pointing overhead. “But they’re not approaching the island. There are dozens upon dozens flying around out here."

ch 19:

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“We once assumed,” Nikli said, noticing her attention, “that the last of the lanceryn had died, and the few hordelings we had bred with them were all that remained. Inferior bloodlines, though they give us the ability to negate some applications of Stormlight. Yours is the third larkin we now know to have survived—but the only one that has grown mature enough to return here.”
Chiri-Chiri had settled down on the table, though she watched the three Sleepless and clicked warningly.
“Why . . . did you say she needed to return?” Rysn asked. “Will she grow sick again?”
“Larger greatshells need to bond mandras—you call them luckspren—to keep from crushing themselves to death with their own weight. The mandras of this place are special. Smaller, yet more potent, than the common breeds. It is no simple thing to make a creature as heavy as a lancer—or larkin, as they are now called—fly. Chiri-Chiri will need to return every few years until she is fully grown.”

Edited March 11 by alder24

[Dofurion]

24 minutes ago, alder24 said:

Larger greatshells need to bond mandras—you call them luckspren—to keep from crushing themselves to death with their own weight. The mandras of this place are special. Smaller, yet more potent, than the common breeds. It is no simple thing to make a creature as heavy as a lancer—or larkin, as they are now called—fly. Chiri-Chiri will need to return every few years until she is fully grown.”

I hadn't written down the part about them being more Smaller.
Well, if there is still a possibility that the Starsprens could have similar properties to the Madras, it may be seen with the Tai-na. Although I am more inclined to have their "Souls" be radiant Sprens or failing that Sprens with large amounts of investiture, like Cusicesh

[alder24]

1 hour ago, Dofurion said:

I hadn't written down the part about them being more Smaller.
Well, if there is still a possibility that the Starsprens could have similar properties to the Madras, it may be seen with the Tai-na. Although I am more inclined to have their "Souls" be radiant Sprens or failing that Sprens with large amounts of investiture, like Cusicesh

Yeah, Starspren are unlikely to be Tai-na spren. It seems that their spren are somewhat sapient. I don't think they are Radiant spren, but some other type of near-sapient great spren. Also Rysn knows Luckspren, from her fall from Tai-na. It's possible they are bonded to some other special species of Luckspren. Dawnshard ch 16:

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And then she saw the spren.
Not the fearspren, but luckspren—like arrowheads with stubby, rippling bodies. They darted through the water around her and Cord. Dozens. Hundreds. Light from the clouded sky above vanished, and Rysn’s ears hurt so much she was forced to equalize by blowing with her nose pinched.
But those spren were glowing, lighting the way, urging them forward.
I know you, spren, she thought. She should have panicked, should have worried about drowning. Instead she watched the spren. How did I fall from so high and not die? Everyone called it a miracle. . . .

Or maybe they are the same type of Luckspren. There are many corpses of greatshells scattered around the coast of Aimia, it might be a part of Tai-na lifecycle to visit Akinah and bond with a special Luckspren, just like other, smaller greatshell come there to die. Tai-na, just like Larkins, are sapient, just like them they require a special amount of assistance because of their enormous size - it would make sense for them both to bond the same spren. Plus it was Tai-na who took care of Larkins after the Scouring of Aima and they can even command them, as evident with Chiri-Chiri. Larkins are also called cousins of Tai-na. There is a connection between Aimia, Larkins and Tai-na. Dawnshard ch 1:

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Talik reached to carefully touch Chiri-Chiri along the top of her head. “So big . . .” he whispered. “I had not realized.”
“What do you mean?” Rysn asked.
“When Aimia fell,” he explained, “the Na-Alind—a family among the greatshelled gods of the Reshi—took in the last of the larkin. Greatshells do not think or speak like people do, and the ways of our gods are strange. But best we can tell, there was a promise among them. To protect these, their cousins.