It probably does not have a IANA assigned BCP 47 language tag, see Codes for constructed languages - Wikipedia , otherwise it would be easy and you could just format a character using that code entered in the Language field of the Font dialog. The language name stated in the status bar then would also appear in the AutoCorrect list.
If not, the easiest would be to use a “reserved for local use” ISO 639-2 code like qcl (all qaa to qtz are reserved for local use, the cl here just for “constructed language”). But such code is completely unspecific and if you plan to share documents is not the proper way to go, as it wouldn’t be local anymore and could clash with other uses.
Rather think about an art-x-privuse tag, where art is the assigned code for “artificial” and x-privuse a private use subtag, where privuse can be any combination of a-z characters up to 8 characters. You should also consult ConLang Code Registry to make sure it doesn’t clash with an already existing definition. Note that the qaa to qtz codes listed there should not be used for the reasons I gave.
LibreOffice does not allow entering arbitrary private-use tags in the character font language field to prevent shooting oneself into the foot… The way to go then would be to edit a stored empty document or template with any language and replace the style’s attribution, i.e. in its content.xml change
<style:text-properties fo:language="en" fo:country="US"/>
(if language was English (USA)) to
<style:text-properties style:rfc-language-tag="art-x-yourconl"/>
and freshen the zip’s content.xml with the change and then reload the document in Writer. The language would then be shown as art (Private-Use=yourconl) {art-x-yourconl} and also be available in AutoCorrect. You just needed to load the empty template to create content attributed with art-x-yourconl, otherwise the language won’t be available.
I or someone else here could help you with that content.xml change if you don’t know how, once you made up your mind what actually to use for yourconl.