Your experience is pretty much what is to be expected with sulfur. You're not doing anything wrong - it's just the nature.of that particular material.

I would suggest keeping pets away from the roses. Here is what Poison Control suggested to a pet (dog) owner:.

"Depending on the concentration and circumstances of exposure, sulfur can be an irritant to a petâÂÂs mucous membranes and skin. If it is ingested, however, sulfur can convert to sulfate in the gastrointestinal tract; sulfate has the potential to act like an acid and cause significant irritation - or even ulceration in severe cases as well as central nervous system or cardiac effects. Because of this, we would recommend keeping your Shih Tzu away from areas of your lawn treated with sulfur until the product has dissipated or has been thoroughly watered"

I have a suggestion for #1: use less. When I used wettable sulfur, I used 3 TB. When I switched to micronized sulfur, I used only 2 TB and got equal results. Be sure to use enough surfactant that it coats the leaves evenly, producing fewer spots. I am not sure what is the best product. I used a little Safer's or Dr. Bronner's soap. Start with a spoonful and see if the sulfur solution flows and sticks to all but the shiniest new leaves. If needed, add another spoonful and test again until you have discovered the minimum effective amount given your water chemistry. If you try a glycol spreader-sticker, measure carefully, because I burned roses with an overdose one time and was afraid to try again.

Suggestion for #2: crank up a boom box and wiggle away.

Oh no, I didn't realize how sulfur might affect my cats and dog. I thought it was nontoxic to mammals! I did more research, and saw this in the Merck Veterinary Manual:

Sulfur and lime-sulfur are 2 of the oldest insecticides. Elemental sulfur is practically devoid of toxicity, although poisoning has occurred occasionally when large amounts were mixed in cattle feed. Specific toxic dosages are not known but probably exceed 4 g/kg. Lime-sulfur, which is a complex of sulfides, may cause irritation, discomfort, or blistering but rarely causes death. Treatment consists of removing residual material and applying bland protective ointments plus any supportive measures that may be indicated.

What fungicide would be safer for my animals? Sulfur? Or tebuconazole (Bayer Advanced Disease Control)? Does tebuconazole also leave a heavy film that can rub off on the animals' fur and get ingested during grooming?

The source you quoted correctly says "Elemental sulfur is practically devoid of toxicity" [to mammals]. People used to swallow a spoonful of sulfur as a spring tonic laxative. If a cat should lick some off her fur, it would do no harm. Perhaps some particularly delicate dogs are allergic to elemental sulfur and various other substances.

Lime sulfur is a completely different chemical that stinks and burns the skin. It was traditionally used as a dormant spray for roses and orchard fruits.

Tebuconazole has a low acute toxicity to mammals, birds, worms, and insects. It does not leave a heavy film on the foliage, but I would keep pets away while it dries. I'd say both tebuconazole and plain sulfur are safe for your animals.

The MSDS for the Bayer Advanced Garden Disease Control for Roses, Flowers & Shrubs Concentrate gives a 1,2-Propylene glycol concentration as 10% by weight.
This product is NOT the normal antifreeze that is so dangerous to cats. However, I suggest that you check if it would attract cats to lick the wet leaves. In that case it would also be licking the wet tebuconazole.

Here is a link that might be useful: MSDS for Bayer rose spray

The Material Safety Data Sheet (MSDS) for Propylene glycol is given below.

Here is a link that might be useful: (MSDS) for Propylene glycol

Propylene glycol is the spreader-sticker in Bayer disease Control.

Henry, do you suppose that propylene glycol diluted to 1 part in 1600 would have enough taste to attract the interest of cats or dogs? I would think not.

I said: "However, I suggest that you check if it would attract cats to lick the wet leaves. In that case it would also be licking the wet tebuconazole."

H.Kuska comment. II would expect that the water part would evaporate much, much faster than the Propylene glycol part. Thus, at one point the dilution could be much smaller. My thinking is "rather safe than sorry".

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"It is a colorless, nearly odorless, clear, viscous liquid with a faintly sweet taste,"
https://en.wikipedia.org/wiki/Propylene_glycol#cite_note-34

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PROPYLENE GLYCOL AND CATS see:
"http://www.lyondellbasell.com/techlit/techlit/2275.pdf

Here is a link that might be useful: PROPYLENE GLYCOL AND CATS

This scientific article reports that ". Cats fed PG had a dose-related increase in Heinz bodies within 2 weeks, and the increase persisted throughout the study."

"Mean RBC survival was decreased in the groups fed 6 or 12% PG by 30 and 55%, respectively, compared with the control group."

H. Kuska comment: Although they only had three sets of data 0%, 6%, and 12%, the finding of a dose dependence and persistence of damage may be important as it suggests (to me) that exposure to lower doses for a longer time period may be of concern regarding the long term health of the cat.

Here is a link that might be useful: research paper

In my first reply I should have been more emphatic about this point: "micronized" sulfur is more effective than "wettable" sulfur and so you can use less to get the same result. Micronized is finer and so gives more thorough coverage. "Garden sulfur" that doesn't specify "wettable" is the coarse grade and is unsuitable for spraying. It can be used to acidify soil or repel ticks and chiggers. This grade also might be labelled "flowers of sulfur."