Distillation: temperature not reaching azeotrope boiling point
I think your major mistake is using a simple distillation method, which may not give accurate reading for azeotropic distillation. The early distillate may not have enough cooling to get to expected azeotrope temperature of aqueous $\ce{HBr}$. If you have followed the work of Carriére et Cerveau (Ref.1), you may able to assume approximate composition of your fractions. Although their work cannot be regarded as of high precision as the composition was determined by titration (Ref.2), they have given an insight to $\ce{HBr}$ distillation. Their work has clearly showed aqueous $\ce{HBr}$ has been yielding a "high-boiling" azeotrope (boiling temperature of azeotrope is higher than boiling points of either of pure components). For example, the $\ce{HBr}$ composition of the early distillate collected at $\pu{110 ^\circ C}$ was reported as 50.8%. The temperature had increased to $\pu{126 ^\circ C}$ maximum (47.5% $\ce{HBr}$), and then declined passed  $\pu{111.5 ^\circ C}$, at which, the $\ce{HBr}$ composition was found to be 30.8%. Since your initial $\ce{HBr}$ concentration was around 30%, I’d say your distillate is already at downward slope and never have able to reach constant boiling azeotrope temperature. If you had used fractional distillation setup with long enough column (and measured atmospheric pressure accurately), you may have able to collect some azeotrope. For your convenience, I listed published data of azeotrope boiling point, density, and composition at different pressure (measured by pressure control apparatus):

$$
\begin{array}{ccc}
\text{Reference} &amp;\text{Pressure} &amp; \text{Boiling Point} &amp; \text{Density at}\: \pu{25 ^\circ C} &amp; \text{Composition, % HBr}\\\hline
\text{Carriére et Cerveau}^1 &amp; \pu{760 mmHg} &amp; \pu{126 ^\circ C} &amp; - &amp; 47.5\\
\text{Ewing and Shadduck}^3 &amp; \pu{760 mmHg} &amp; \pu{125 ^\circ C} &amp; - &amp; 47.795 \pm0.033\\
\text{Bonner et al.}^2 &amp; \pu{760\pm0.5 mmHg} &amp; \pu{124.3\pm0.02 ^\circ C} &amp; - &amp; 47.63\pm0.01\\
\text{Bonner et al.}^2 &amp; \pu{100 mmHg} &amp; \pu{74.12 ^\circ C} &amp; 1.5116 &amp; 49.80\\
\text{Bonner et al.}^2 &amp; \pu{200 mmHg} &amp; \pu{90.35 ^\circ C} &amp; 1.5030 &amp; 49.28\\
\text{Bonner et al.}^2 &amp; \pu{300 mmHg} &amp; \pu{99.91 ^\circ C} &amp; 1.4961 &amp; 48.83\\
\text{Bonner et al.}^2 &amp; \pu{400 mmHg} &amp; \pu{107.00 ^\circ C} &amp; 1.4908 &amp; 48.47\\
\text{Bonner et al.}^2 &amp; \pu{500 mmHg} &amp; \pu{112.94 ^\circ C} &amp; 1.4866 &amp; 48.19\\
\text{Bonner et al.}^2 &amp; \pu{600 mmHg} &amp; \pu{117.82 ^\circ C} &amp; 1.4832 &amp; 47.95\\
\text{Bonner et al.}^2 &amp; \pu{700 mmHg} &amp; \pu{122.00 ^\circ C} &amp; 1.4802 &amp; 47.74\\
\text{Bonner et al.}^2 &amp; \pu{800 mmHg} &amp; \pu{125.79 ^\circ C} &amp; 1.4775 &amp; 47.56\\
\text{Bonner et al.}^2 &amp; \pu{900 mmHg} &amp; \pu{129.13 ^\circ C} &amp; 1.4752 &amp; 47.40\\
\text{Bonner et al.}^2 &amp; \pu{1000 mmHg} &amp; \pu{132.12 ^\circ C} &amp; 1.4733 &amp; 47.27\\
\text{Bonner et al.}^2 &amp; \pu{1100 mmHg} &amp; \pu{134.80 ^\circ C} &amp; 1.4716 &amp; 47.14\\
\text{Bonner et al.}^2 &amp; \pu{1200 mmHg} &amp; \pu{137.34 ^\circ C} &amp; 1.4700 &amp; 47.03\\
\end{array}
$$



References:


MM. E. Carriére and Cerveau, Compt. rend. Acad. Sci. 1923, 177, 46-48 (https://gallica.bnf.fr/ark:/12148/bpt6k3130n/f50.image).
W. D. Bonner, L. G. Bonner, F. J. Gurney, “Azeotropic Hydrobromic Acid Solutions at Pressures of 100 mm to 1200 mm,” J. Am. Chem. Soc. 1933, 55(4), 1406-1409 (DOI: 10.1021/ja01331a012).
D. T. Ewing, H. A. Shadduck, “The Composition of a Constant-Boiling Solution of Hydrogen Bromide in Water,” J. Am. Chem. Soc. 1925, 47(7), 1901–1904 (DOI: 10.1021/ja01684a017).