Converting/transcoding AAC audio to blu-ray compliance format, with minimal quality loss?

How would I go about converting/transcoding AAC audio to a Blu-ray compliant format, with little to no quality loss?

I am specifically looking to convert/transcode the audio from files, I have with the following technical specs:

[MediaInfo info:]
MediaInfo
General
Unique ID: 188898217902865359047639781233872137384 (0x8E1C7932B63B776B8C434B0619A4ECA8)
Format: Matroska
Format version: Version 4 / Version 2
File size: 864 MiB
Duration: 21mn 38s
Overall bit rate: 5 585 Kbps
Encoded date: UTC 2017-07-19 07:15:32
Writing application: mkvmerge v8.3.0 (‘Over the Horizon’) 64bit
Writing Library: libebml v1.3.1 + libmatroska v1.4.2

Video
ID: 1
Format: AVC
Format/Info: Advanced Video Codec
Format profile: High@L4
Format settings, CABAC: Yes
Format settings, ReFrames: 2 frames
Codec ID: V_MPEG4/ISO/AVC
Duration: 21mn 38s
Bit rate: 5 327 Kbps
Width: 1 920 pixels
Height: 1 080 pixels
Display aspect ratio: 16:9
Frame rate mode: Constant
Frame rate: 23.976 (24000/1001) fps
Color space: YUV
Chroma subsampling: 4:2:0
Bit depth: 8 bits
Scan type: Progressive
Bits/(Pixel*Frame): 0.107
Stream size: 824 MiB (95%)
Language: English
Default: Yes
Forced: No
Color range: Limited
Color primaries: BT.709
Transfer characteristics: BT.709
Matrix coefficients: BT.709

Audio
ID: 2
Format: AAC
Format/Info: Advanced Audio Codec
Format profile: LC
Codec ID: A_AAC
Duratio : 21mn 38s
Bit rate: 256 Kbps
Channel(s): 2 channels
Channel positions: Front: L R
Sampling rat : 44.1 KHz
Frame rate: 43.066 fps (1024 spf)
Compression mode: Lossy
Stream size: 39.6 MiB (5%)
Language: English
Default: Yes
Forced: No

Any help would be much appreciated!
I am looking to make my own custom BD disc with the files!
Thanks on advance for your help!