There's a simple answer - for trains through the tunnel the relevant WTT will be a Eurotunnel document with the Eurostar and shuttle paths in it, in addition to freight to and from Dollands Moor. Realtime trains or similar websites don't show these WTT paths and they don't exist as schedules in TRUST (except for Eurostar trains, which obviously run over Network Rail infrastructure on the CTRL).
I don't think the Eurotunnel WTT is a public document, but there is some technical documentation on the Eurotunnel website (the infrastructure operator "Groupe Eurotunnel" is now known as Getlink) going into the details of the access conditions, timetabling process and charges for using the tunnel. PDF documents titled "Fixed Link Usage Annual Statement" for 2022 to 2024 are available from this page:
https://www.getlinkgroup.com/en/our-group/eurotunnel/eurotunnel-railway-network/
For example it says this about the general timetabling process and allocation of paths:
Schedules for freight within the tunnel don't really concern Network Rail too much, as they only briefly touch Network Rail infrastructure next to the Eurotunnel boundary on their journey to or from Dollands Moor Yard. For the same reason, you have Eurostar schedules along the CTRL that have a timing point at the Eurotunnel boundary and then nothing until Paris or Brussels.
The VSTP tunnel schedules that appear in TRUST (and therefore Realtime trains and so on) are created by the staff at Dollands Moor Yard, mainly for the purpose of allowing a TOPS wagon consist to be input for tunnel freight, which in turn allows visibility on TOPS of wagons being in the UK or not. Once wagons arrive at Frethun yard they are automatically shown as being "off TOPS" with a description of "TO CONTINENT." Internationally registered wagons which left the UK years ago, never to return, can sometimes be found in TOPS with a last reported location of Dunkerque, arriving on a date prior to the end of train ferry operations (the ferry had regular schedules created for it like a train and was probably in the British Rail WTT somewhere).
GBRF (who have a separate DOLMGBF location for Dollands Moor schedules) don't work quite the same way, for example the Antwerp - Irvine china clay service (which has now ceased) was not normally entered onto any schedule through the tunnel, since the control of trains within the tunnel is not based on TRUST. On Network Rail, TRUST schedules are firmly linked to control of trains, signalling and track access but this doesn't apply to Eurotunnel, so DB tunnel schedules are entered as required, based on what has already been agreed with Eurotunnel either ad hoc or using an existing WTT path.
The regular tunnel freight services will have WTT slots (and regular headcodes) but given the obvious potential for late-running of trains coming long distances over international borders, the actual time they run may differ on the day. One consideration is that it is obviously preferable to take a train to France and bring one back, rather than needing to run light engine in one direction, another is that there would probably be more freight paths available during the night, depending on engineering work. Light engine paths through the tunnel can be distinguished by the second digit of the headcode (tunnel headcodes don't have letters in them) being a 5, such as 4501. From what I can remember reading somewhere, I think Eurotunnel actually refer to trains by a 5-digit headcode similar to continental practice, but only 4 digits appear in TRUST.
Back when there was more channel tunnel freight, in the era of Railfreight Distribution for example, it's quite possible that tunnel trains appeared in UK WTT documentation as they had to be carefully co-ordinated with SNCF. Schedules to Dollands Moor still have a UIC code associated with them which is (or at least was) intended to be the 5-digit European headcode for the train's onward journey into Europe (e.g. 47414). These days, with a handful of distinct regular freight flows operating through the tunnel, there is less potential for confusion.
There's a Cinerail DVD (originally released on VHS) available for £10 which gives a good historical insight into tunnel freight operations back in the days when pairs of SNCF electrics worked through the tunnel:
https://www.videoscene.co.uk/the-channel-tunnel-gateway-to-europe