From my giant list of ternary genera with spinor genera and list of sporadic numbers up to some bound I don't remember.
As Jeremy points out, this one has bad features, not spinor regular, more than one squareclass of numbers missing compared with the full genus. It might be possible to prove good behavior with local restriction such as $8k+1,$ but that misses your desired $8k+3$ Let me see what Kap article you've cited.
http://zakuski.math.utsa.edu/~jagy/report_1000.txt
$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$
=====Discriminant 256 ==Genus Size== 2
Discriminant 256
Spinor genus misses no exceptions
256: 1 2 32 0 0 0 vs. s.g. regular candidate
256: 2 4 9 4 0 0 vs. s.g. 1 3 43 163 907
--------------------------size 2
The 150 smallest numbers represented by full genus
1 2 3 4 6 8 9 11 12 16
17 18 19 22 24 25 27 32 33 34
35 36 38 40 41 43 44 48 49 50
51 54 56 57 59 64 65 66 67 68
70 72 73 75 76 80 81 82 83 86
88 89 91 96 97 98 99 100 102 104
105 107 108 113 114 115 118 120 121 123
128 129 130 131 132 134 136 137 139 140
144 145 146 147 150 152 153 155 160 161
162 163 164 166 168 169 171 172 176 177
178 179 182 184 185 187 192 193 194 195
196 198 200 201 203 204 208 209 210 211
214 216 217 219 224 225 226 227 228 230
232 233 235 236 241 242 243 246 248 249
251 256 257 258 259 260 262 264 265 267
The 150 smallest numbers NOT represented by full genus
5 7 10 13 14 15 20 21 23 26
28 29 30 31 37 39 42 45 46 47
52 53 55 58 60 61 62 63 69 71
74 77 78 79 84 85 87 90 92 93
94 95 101 103 106 109 110 111 112 116
117 119 122 124 125 126 127 133 135 138
141 142 143 148 149 151 154 156 157 158
159 165 167 170 173 174 175 180 181 183
186 188 189 190 191 197 199 202 205 206
207 212 213 215 218 220 221 222 223 229
231 234 237 238 239 240 244 245 247 250
252 253 254 255 261 263 266 269 270 271
276 277 279 282 284 285 286 287 293 295
298 301 302 303 308 309 311 314 316 317
318 319 325 327 330 333 334 335 340 341
Disc: 256
==================================
256: 1 2 32 0 0 0
misses, compared with full genus
256: 2 4 9 4 0 0
misses, compared with full genus
1: 1 3 43 163 907
----------------------------------------
$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$