Can I engrave borosilicate glassware without compromising its structural integrity?
Any scratch weakens glass somewhat. However, a vibratory engraver is very likely to shatter the vessel.
You can label glass permanently by etching with warm concentrated $\ce{NaOH}$ or room-tempeerature $\ce{HF}$, perhaps with a stencil. Use great caution with $\ce{NaOH}$, and even more caution with $\ce{HF}$. Both are very caustic, and $\ce{HF}$ can release toxic fumes and cause deep tissue damage.
There are commercial etching creams with HF that might be less likely to splash.
Note that etching weakens glass slightly, but less so then engraving.