when you are re-projecting (only on the same horizontal datum!!!) then you also need to set the vertical datum again (for the few datums that are supported). It seems you have an older version of LAStools because the NN2000 is only supported in more recent versions.
Note that the GeoTIFF tag 4097 is merely referring to a textual (human-readable) string that spells out the name of the vertical datum. The relevant (machine-readable) GeoTIFF tag is 4096 and the value 5941 for NN2000 was not supported in your older version of LAStools.
Below is an example (that makes no actual sense) of how to use the relatively new option '-vertical_nn2000' that adds GeoTIFF tag 4096 with value 5941 to specify NN2000.
E:\LAStools\bin>las2las -i ..\data\fusa.laz ^
-target_epsg 25832 ^
-vertical_nn2000 ^
-o poopypoop.laz
E:\LAStools\bin>lasinfo poopypoop.laz
lasinfo (160921) report for poopypoop.laz
reporting all LAS header entries:
file signature: 'LASF'
file source ID: 0
global_encoding: 0
project ID GUID data 1-4: 00000000-0000-0000-0000-000000000000
version major.minor: 1.1
system identifier: 'LAStools (c) by rapidlasso GmbH'
generating software: 'las2las (version 161004)'
file creation day/year: 40/2010
header size: 227
offset to point data: 329
number var. length records: 1
point data format: 1
point data record length: 28
number of point records: 277573
number of points by return: 263413 13879 281 0 0
scale factor x y z: 0.01 0.01 0.01
offset x y z: 11500000 -19800000 0
min x y z: 11547909.99 -19854595.57 42.21
max x y z: 11550165.02 -19850503.42 64.35
variable length header record 1 of 1:
reserved 0