What i can tell from the maps is:
On 8th September LSSAH was just north of Beryslav, aproaching the german bridge that connects Beryslav with the Kakhovka bridgehead. Surely the germans had some kind of ponton bridge or ferries.
Unfortunately i lack the map of 9th September. The original files can be up to 100mb for one map and download speed from the FTP server is pretty slow, so i am not able to provide day-by-day informations yet. I already dowloaded more than 200 situation maps, but it will take me some more time to get them all...
On 10th September we have the following german units in the bridgehead from west to east:
73. ID, A.A. LSSAH
, 46. ID, V.A. XXX Korps, 22. ID, 170. ID, 1. Mountain, V.A. XXXXIX Korps. The 72. ID is crossing the pontoon bridge, LSSAH still on the northern side. So that day the A.A. of LSSAH is already fighting south of the Dnieper. If you got the information that parts of LSSAH were fighting in the bridgehead on 9th September, it was probably this Aufklärungsabteilung
On 11th September, the bulk of the LSSAH is crossing the pontoon bridge while parts (Tle.) of the divsion are fighting east of the brisgehead. The A.A., together with the V.A. of LIV Korps are making a speedy approach towards Perekop.
I read on the internet that they reached the outskirts? of Perekop, but got repelled and the raid failed.
On 15th September the bulk of LSSAH is making a speedy approach towards the Chonhar Pensinula/Chongarsky Isthmus, probably they tried again to break through in a coup, but got repelled and then moved towards Melitopol. Some parts of the division were still clearing the area southwest of Kherson.
On 25th September the division is again in a position north of Perekop.
On 29th September the division is in the very north of the german front, just south of the big Dnieper bend.
On 3rd/4th October the division is few kilometers northwest of Melitopol.
On 7th October LSSAH is approaching Berdjansk, meeting with the 16th and 14th Panzer Divisions, closing the pocket.
On 8th October LSSAH captured Mariupol.
Sometimes divisions make rapid movements left and right and back and forth and i ask myself it it can be true, but on the other hand i am pretty sure the general staff had accurate maps at hand
At least the positions of the own units should be accurate
For now, i'll stop here. Thats what i can tell when looking at the maps. You see, many days are still missing in my map collection...
If you want to know about later dates, please feel free to ask
Map of 11th September 1941: