This article That I found seems to address your question specifically.  The solutions are pet friendly, but sometimes specific to the beetle grub in your lawn.  If you take your grub to your county extension agent, specific ID may be possible.

https://www.gardenguides.com/info_8114217_pet-safe-lawn-grub-control.html

Pet Safe Lawn Grub Control

By: G.D. Palmer

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21 September, 2017

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White lawn grubs, the larvae of scarab beetle species like the June beetle, Japanese beetle and European chafer, are a lawn pests during the summer months. These grubs live under the soil, feeding on grass and other plant roots. They cause wilting and browning of the lawn, especially in the late summer months. Many products are available for grub control, but not all are safe for use around pets or small children.  Dry conditions in midsummer often cause lawns to brown and go dormant naturally. This condition may be unattractive, but it also discourages beetles from laying their eggs in the grass. Once rainfall resumes, the dormant lawn becomes green again without a high risk of grub damage. Letting your lawn go dormant costs nothing and involves no chemicals or harmful products. This technique isn't reliable in areas with higher summer rainfall. Homeowners living in communities with rules about their lawn's appearance may also be unable to use dormancy as a grub control method.

Nematodes
Nematodes, tiny wormlike creatures living in the soil, are used to control grubs without risk to humans or animals. Heterorhabditis bacteriophora nematodes enter the grub and release bacteria that kills it. Often sold as Hb nematodes, apply this product late in the day when the soil is moist. Water the lawn immediately after application to activate the nematodes. Nematodes may take some time to work and aren't always as effective as chemical methods. However, these natural grub-killers won't hurt your pets.

Bacterial Milky Disease

This bacteria control only Japanese beetle grubs and is ineffective against other lawn grub species. It attacks and kills the larvae, preventing the beetles from successfully reproducing. Bacterial milky disease is a slow-acting control method and requires 3 to 5 years to become established, according to Pennsylvania State University. This bacteria attacks only beetles and will not harm humans or animals.

Imidicloprid

This insect-growth inhibitor is also used in cat and dog spot-on flea treatments, making it one of the less-dangerous grub-control chemicals for pets. Imidicloprid is a preventive product that kills only newly hatched grubs. Apply imidicloprid products during June and July and the water it in deeply to kill the late summer hatching. Like other pesticides, imidicloprid can be dangerous to pets in large quantities or when ingested. Keep pets off the lawn for a few days after applying the grub treatment, especially kittens, puppies or pregnant animals.

I hope this info will help you make the best decision for your particular situation.  If I can help more, please do not hesitate to contact me.

you are welcome.