1. First clue: electronic configuration is not affected by the nature of ligand

In coordination chemistry, this means the metal ion has a configuration for which ligand field splitting does not change the electron arrangement.

This happens for ions with:

• $d^0$
• $d^1$
• $d^5$ high spin
• $d^{10}$

Among the given ions:

• $\mathrm{Cr}^{3+}$: $[Ar]3d^3$
• $\mathrm{Ti}^{3+}$: $[Ar]3d^1$
• $\mathrm{Ni}^{2+}$: $[Ar]3d^8$
• $\mathrm{Mn}^{2+}$: $[Ar]3d^5$

So the ions whose configuration is essentially unaffected by ligand nature are:

• $\mathrm{Ti}^{3+}$ ($d^1$)
• $\mathrm{Mn}^{2+}$ ($d^5$)

Thus, from the first clue alone, options B and D remain possible.

2. Second clue: gives violet colour in non-luminous flame under hot condition in borax bead test

Recall standard borax bead test colours:

• $\mathrm{Cr}^{3+}$: green bead
• $\mathrm{Ni}^{2+}$: brown bead in oxidizing flame, grey in reducing flame
• $\mathrm{Mn}^{2+}$: amethyst/violet bead in oxidizing flame, but becomes colourless in reducing flame
• $\mathrm{Ti}^{3+}$: violet bead under reducing/hot conditions

The question says violet colour in non-luminous flame under hot condition. This is characteristic of titanium.

So the ion is $\mathrm{Ti}^{3+}$.

3. Check each option

• A. $\mathrm{Cr}^{3+}$: $d^3$; not the relevant unaffected configuration, and borax bead is green. Incorrect.
• B. $\mathrm{Ti}^{3+}$: $d^1$, configuration unaffected by ligand nature; gives violet colour in hot borax bead. Correct.
• C. $\mathrm{Ni}^{2+}$: $d^8$, configuration affected by ligand field; borax bead not violet. Incorrect.
• D. $\mathrm{Mn}^{2+}$: $d^5$, ligand-independent high-spin case, but borax bead test classically gives amethyst in oxidizing flame, not the described hot violet bead. Incorrect.

4. Final answer

The correct option is:

$\boxed{\mathrm{Ti}^{3+}}$

So, Option B is correct.