I don't know if the K prefix guns shared the same chronological order of serialization along with other TC muzzle loaders or not.
But I'm assuming that they do.
Also, some folks have reported serial numbers for guns and kit guns that no one really knows how long after production that each was purchased.
My guess is that a kit gun may have sat on the shelf for 3-6 years after production, more or less.
Based on that criteria, my educated guess would put the production of your gun in the mid to later half of the 1980's I think.
But that really depends how long the average TC Hawken or Renegade kit sat in the warehouse or store shelf waiting to be sold.
That could account for some overlap with the dates reported below.
I found a post in a thread about TC serial numbers stating: --->>>
https://www.muzzleloadingforum.com/threads/serial-number-dating-a-t-c.34284/page-8#post-1639079
Thompson Center .50 Hawken, bought from Walmart about 1986 as a kit. s/n K3502XX
Another post about 2 Renegade serial numbers, one was a kit: --->>>
https://www.muzzleloadingforum.com/threads/serial-number-dating-a-t-c.34284/#post-387248
Renegade 54, bought new in 1992, #319XXX
Renegade 50, bought new in 1993, #353XXX (kit)
Lastly: --->>>
https://www.muzzleloadingforum.com/threads/serial-number-dating-a-t-c.34284/#post-387545
Thompson Center Renegade .54 Cal #357xxx, Bought in 1992 as a kit.
Here's some additional anecdotal info.:
Other folks here on the forum who have posted when they bought there "new" Renegades puts 5-digit as starting about 1978/79 and 6-digit starting about 1981/82 on the Renegades (with 1992/93 rifles being in the 350XXX serial range). Speculation for sure, but its about as close as can be guessed with no support from TC themselves. --->>>
https://www.muzzleloadingforum.com/...r-warning-to-their-barrels.52617/#post-625088