Just remove the parameter limits and things are fine. In particular limits on height and position are not useful because you have from TSpectrum good starting values:
FCN=102635 FROM MIGRAD STATUS=CONVERGED 20377 CALLS 20378 TOTAL
EDM=3.01172e-07 STRATEGY= 1 ERROR MATRIX UNCERTAINTY 1.2 per cent
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 1.02541e+04 9.31222e+01 2.93468e-02 -4.00450e-06
2 p1 1.02369e+02 1.56045e-02 -1.88708e-06 -9.60459e-04
3 p2 2.08880e+00 4.05840e-02 -8.57820e-06 -8.86939e-03
4 p3 1.13093e+00 1.47663e-02 -3.14096e-06 -1.90742e-02
5 p4 1.98080e+04 6.90166e+01 4.96109e-02 4.37922e-06
6 p5 1.60307e+02 1.62176e-02 -5.24457e-06 -9.25648e-03
7 p6 5.62502e+00 3.74982e-02 -2.81694e-05 2.71375e-02
8 p7 1.15677e+00 5.26796e-03 -3.22768e-06 -7.56588e-02
9 p8 5.24787e+04 1.27003e+02 -3.03361e-02 9.13087e-08
10 p9 2.31313e+02 9.33004e-03 5.11277e-07 -1.21506e-02
11 p10 4.38126e+00 2.09624e-02 4.01907e-06 7.59486e-04
12 p11 1.03412e+00 2.93985e-03 3.58288e-07 -2.21235e-02
13 p12 7.42729e+04 1.41314e+02 -1.89999e-02 3.08412e-06
14 p13 3.02322e+02 8.46518e-03 -8.79842e-07 6.41160e-03
15 p14 5.19092e+00 1.91111e-02 7.90502e-07 4.33323e-02
16 p15 1.07028e+00 2.44471e-03 3.15843e-07 -2.18970e-01
17 p16 8.49393e+04 1.49020e+02 -5.99618e-02 1.72516e-06
18 p17 3.74984e+02 8.33811e-03 2.29657e-06 -2.24234e-02
19 p18 5.80169e+00 1.97632e-02 8.05751e-06 -3.28313e-03
20 p19 1.08514e+00 2.37760e-03 1.01951e-06 2.63056e-02
21 p20 8.05968e+04 1.37781e+02 -6.14525e-02 -8.81422e-07
22 p21 4.47406e+02 9.37127e-03 -2.44565e-06 2.43563e-03
23 p22 6.57894e+00 2.12416e-02 1.43575e-05 -7.23418e-02
24 p23 1.13316e+00 2.53259e-03 1.16047e-06 5.32616e-01
25 p24 6.67695e+04 1.21475e+02 1.83583e-02 -3.98217e-07
26 p25 5.19249e+02 1.05880e-02 1.64783e-06 8.87442e-03
27 p26 7.29428e+00 2.51076e-02 3.62715e-06 -1.97886e-02
28 p27 1.15999e+00 2.90617e-03 1.37021e-07 -8.94262e-02
29 p28 4.87018e+04 9.61724e+01 -3.38996e-02 2.92026e-06
30 p29 5.90926e+02 1.31483e-02 7.29729e-06 -2.48014e-02
31 p30 8.18759e+00 2.98714e-02 1.28891e-05 2.25773e-02
32 p31 1.20286e+00 3.34548e-03 1.48882e-06 2.00579e-01
33 p32 3.21473e+04 6.23603e+01 -3.39265e-03 -3.72801e-07
34 p33 6.62491e+02 1.82036e-02 -5.95112e-06 7.59008e-03
35 p34 9.23128e+00 2.92409e-02 1.96502e-05 -1.05662e-02
36 p35 1.25903e+00 3.19849e-03 2.26718e-06 -6.72207e-01
37 p36 1.96069e+04 5.75669e+01 -5.28892e-02 -2.48866e-08
38 p37 7.33769e+02 2.25002e-02 -1.70649e-05 1.67897e-03
39 p38 1.02151e+01 5.46677e-02 4.02298e-05 1.02870e-03
40 p39 1.30150e+00 6.78401e-03 4.39096e-06 1.22155e-01
41 p40 1.11834e+04 4.10681e+01 -1.29588e-02 -7.19223e-06
42 p41 8.04644e+02 3.41669e-02 1.80084e-05 -4.23791e-03
43 p42 1.12243e+01 6.84486e-02 3.45098e-05 -3.25288e-03
44 p43 1.37001e+00 9.62359e-03 2.77547e-07 2.80403e-01
45 p44 6.16709e+03 3.28097e+01 -2.74448e-02 -6.31651e-07
46 p45 8.75303e+02 4.92328e-02 -2.02788e-05 -2.68062e-03
47 p46 1.17185e+01 1.10730e-01 7.49818e-05 -4.07167e-03
48 p47 1.34753e+00 1.51052e-02 1.45881e-05 3.29182e-02
49 p48 3.05099e+03 2.22991e+01 3.83340e-04 -5.99980e-06
50 p49 9.45659e+02 7.90957e-02 2.96576e-05 -1.55813e-03
51 p50 1.31745e+01 1.53065e-01 -2.36602e-05 -4.78300e-03
52 p51 1.48086e+00 2.43069e-02 -1.90073e-05 3.59031e-02
53 p52 1.52680e+03 1.63800e+01 -1.08539e-02 -8.37003e-06
54 p53 1.01593e+03 1.18765e-01 -7.44742e-05 1.84926e-03
55 p54 1.35733e+01 2.43258e-01 8.14097e-05 -2.95641e-03
56 p55 1.39966e+00 3.95722e-02 2.52129e-05 2.40325e-02
57 p56 6.86886e+02 1.49710e+01 1.05628e-02 -2.97634e-05
58 p57 1.08558e+03 2.09438e-01 -3.51877e-05 6.89297e-04
59 p58 1.42634e+01 3.32508e-01 9.26532e-05 -1.27887e-03
60 p59 1.56719e+00 6.97112e-02 -4.16264e-05 7.68160e-03
61 p60 3.34848e+02 9.80470e+00 -4.87618e-03 -4.12954e-05
62 p61 1.15472e+03 3.29875e-01 2.07314e-04 -6.31440e-04
63 p62 1.47865e+01 7.32662e-01 2.84881e-04 -5.98310e-04
64 p63 1.28515e+00 1.46152e-01 1.12130e-04 2.17236e-03
The HyperGaussians reduced the chi-squared roughly an order of magnitude but your high-statistics spectrum shows that the shape might need further improvements. Looking at the higher ADC channels, one can see that the peak shape is asymmetric, it is wider on the right side. One could incorporate this by introducing a fifth parameter in the peak shape, the asymmetry: x < peak position: sigma*(1-asym), x > peak position: sigma*(1+asym) .
You have parametrized each peak with its own set of parameters. In order to reduce the
number of fit parameters and stabilize/constrain the fit you might want to describe sigma, gamma and asym as polynomials as a function of the ADC channel.