Thread is gonna be your cleanest option and it's what most people doing detailed models swear by. you want a good quality waxed thread or even fishing line - the waxed stuff handles way better because it doesn't tangle as much and it's easier to keep tension on while you're tying. wire can work but it's messier to deal with, more prone to snapping if you're not careful, and it leaves little metal bits everywhere which is not ideal near your kitchen haha. plus thread looks more historically accurate for something like HMS Victory anyway. grab yourself some tweezers, a small sharp pair of scissors, and honestly a magnifying lamp will save your sanity - trust me on this. the thread comes in spools and doesn't shed like loose materials do.
the real trick to keeping it mess-free is doing prep work first. lay out all your rigging lines on a piece of paper and number them or mark them somehow so you're not constantly referring back to those confusing instructions while fumbling around. measure and cut everything before you start attaching to the model - way less rummaging through bits. keep a small container or cup nearby for your scraps rather than letting them scatter across your kitchen counter. also work with a white or neutral colored mat under your model so you can actually see what you're doing and spot any stray threads immediately.
one thing that helped me with similar projects - test your technique on something cheap first. grab some scrap wood or even just use some bits from the kit itself to practice your knots and attachment methods. once you know what you're doing the actual rigging goes faster and cleaner. and yeah, it's gonna take patience, especially on something intricate like Victory, but thread really is the way to go for minimal mess.