The stick shouldn't affect the sound by much, but how it's "socketed" onto the tub (for that kind of tub) could have a strong influence, and a change in sticks might cause you to change posture. (If you use your foot to hold the tub down, moving it slightly could affect tone.)

Broom handles are sometimes a relatively soft wood, and - as suggested - prone to break. The kind that "screws into" a shop broom will likely be a little harder than one cut off of a "broomstraw" sweeper. If you want a smaller diameter stick, rake handles, or hoe handles, are sometimes a good choice, since they're more often made of harder wood; but you're unlikely to find the kind of wood accurately named on the label of replacement handles. At hardware stores good rake/hoe handles run $20+(US) in my area, although they can be found for half that in "possibly inferior" wood, if you select sound pieces individually.

Closet rod, at 1.25 inches (3 cm) is a good choice if you want a "fatter" stick. I have several "well aged" ones I've used for holding up tarps in the campground when they were cheap, but I found the rod priced at a little over $1.20 (US) per foot the last time I looked for one.

Shovel handles are generally "harder" wood, but often have a bend at the end to fit into the blade. That may be just what you want, or may make them unusable depending on your "tub design." The "long straight" shovel handles seem hard to find in my area, and I'd think the "T-Handle" kind would be too short for most tub use.

Axe handles and mattock handles generally will be made from quite hard woods, but usually have peculiar bends and tapers that may be a problem. They are seldom round.

The C-section handrail would seem to me to have the same "asymmetry problem" but every tub - and every player - presents different oproblems and preferences. Round handrails, usually with at least a small flat on the underside, and other shapes can be found; but you might have to go to "specialty" shops. They'd likely be of good wood.

"Natural" sticks are sometimes found that can be a nice touch, but "fallen limbs" are seldom of sound quality and it may be difficult to find straight enough ones that you can cut from live trees/bushes. You might need to follow a tree-trimmer around for a while to find a really good one. Festivals often have "craft fairs" where you can find people vending "walking sticks" that might offer some interesting possibilities, but most of the regulars in my area make the majority of their sticks too short for my taste even as walkers.

One fellow who's been at our local festival for a couple of years uses a "doubletree"^* for a stick, but he's using a six foot diameter "tub" and 3/16" aircraft cable for a string. (He sweats a lot when he plays; but he's "proud of being loud" - his main talent. I expect to see him with a 300 gallon stock tank and some bridge-structure parts if his "talent" continues to develop.)

^* From Random House web dictionary:
doubletree [duhb-uhl-tree]
–noun
a pivoted bar with a whiffletree attached to each end, used in harnessing two horses abreast.
Origin:
1840–50, Americanism ; modeled on singletree
(All clear now?)

John