I bought a BTD60BK ... has a warning on using it on the T700W model. Would there be any problem if I use the BTD60BK in the DCP-T700W?
If the manufacturer of the ink
specifically states to not use it with your printer model then do not use it with your printer.
As Ink Stained Fingers wrote, one of the black inks may be Dye-based ink and the other a Pigment ink. Print heads are designed to use either Dye-based ink or Pigment ink. While there are some exceptions to the rule, you should only use ink designed for that print head.
Your printer uses 4 ink colors - Cyan, Yellow, Magenta and Black. Inkjet printers with 4 colors only are typically Dye-based inks for CYM and Pigment ink for Black. The Pigment Black is used for text and gives a better blackness, sharper resolution of text and provides some water resistance such as to highlighters. The Dye-based CYM inks are mixed to give black when printing images. Using a Pigment ink in a print head that is designed for Dye-based ink will cause clogs because of clumping of pigment particles and may result in permanent failure of the print head.
Other 4 color printers may use only Pigment Ink for all the colors. You generally would not use a Dye-based ink in pigment ink printers. Though it may not cause clogging, the dye-ink may result in poorer looking images and off-colors.
All that said, it is difficult to tell what kind of ink Brother printers like yours use. I found a Chinese web site offering aftermarket inks
"BTD60BK BT60 BK BT5000 BT5001 C CISS dye ink Refill Kit for brother DCP-T310 DCP-T510W DCP-T710W"
https://www.aliexpress.com/i/32810989794.html
This appears to say that BTD60BK is a Dye-based ink.
Another Chinese web site sells their aftermarket ink for the DCP-T700W, your printer, that states on the bottles it is an all Dye-based ink for all colors.
https://www.aliexpress.com/i/32967146738.html?spm=2114.12057483.0.0.f5c2ab196g1rw3
Then there is another web site selling the OEM Brother ink that explicitly states
" **Note : BTD60BK is not applicable to DCP-T300, DCP-T500W, DCP-T700W and MFC-T800W.** "
and
" **Note : BT6000BK is not applicable to DCP-T310, DCP-T510W, DCP-T710W, MFC-T810W,MFC-T910DW.**"
https://shopee.com.my/Brother-Refil...-BT5000C-BT5000M-BT5000Y-i.41430992.722931082
So what to do?
Maybe both Black inks are Dye-based as the Chinese web sites state. Maybe one of the Chinese web sites is incorrect. Maybe the Black inks are both Dye-based, or both Pigment inks, but their formulations are different enough to either cause problems with the print head or poor results aesthetically or functionally on the printed page.
Until more information is known about these inks, I would heed Brother's warning.
Please let us know what you decide and what happens.