You'd probably have to walk the Tolosana/Arles route from Toulouse to Maubourguet (etapa 27 here on Gronze). The accommodation on the Tolosana is a bit of a hit or miss, even in the high season (for example, you can only get the keys for the communal gîtes on week days). No idea what will be open in October, but Gronze should have some info.
From Maubourguet you walk the GR101 to Lourdes. I don't have any other information about this route, but there might be something on the forum if you do a search!
[p.s. neither of these routes has anything to do with the Piémont]
I haven't walked that way but when I was in Lourdes last year I met folks who had and no one mentioned anything that would suggest it wasn't well marked. As for accommodation I would suspect it similar to the Pyrenees route I was on: there was accommodation but I pretty well always had to phone to get some one to come and unlock the door or tell me which house nearby to knock on the door to get a key. Folks were always helpful.
There is a new GR trail linking Toulouse and Saint Bertrand de Comminges, where you connect with the Piedmont Way and eventually arrive in Lourdes. It has been called via Garona (it goes along this beautiful river), classified as GR861 It is 170 km long, and has been estimated as a 8-10 days walk. See info here
No albergues, so it could be more expensive. On the other side, in the the usual Caminos there are not many chances to be a pioneer, as in the old times. I have not (still) walked it -it is in my bucket list.
Some hiking experience and basic French abilities seem advisable.
Edited: Just discovered this videoabout the Via Garona. You can see, Southwest France (the Occitanie) is amazingly beautiful.
One runs straight south from the city centre on the left bank of the river, up-river, crossing over at Roques to Pinsaguel, then on up river via Muret and Carbonne, eventually Saint-Gaudens, eventually joining the Piémont Way at Barbazan.
The other follows the Arles Way out of Toulouse via Colomiers, Pibraq, Léguevin and so on, L'Isle-Jourdain, Auch, eventually Maubourguet ; there is then a waymarked route from Maubourguet to Lourdes, with two variants -- either straight south from Maubourguet up river along the Adour to Tarbes, then either a little DIY along that same up-river route up to the Piémont Way at Bagnères de Bigorre ; or carry along the Arles Way a little further after Maubourguet, then take the GR101 through a lonelier route directly to Lourdes.
The route via Saint-Gaudens would have many advantages at start until Barbazan, i.e. it goes through many small towns and villages as well as being mostly a river valley route. It also gets you quickest out of the suburbs. Saint-Gaudens is also a lovely town BTW. The Piémont Way between Barbazan and Lourdes is more mountainous, but the waymarked track does occasionally lead via excessive detours up and downhill rather than a quicker small country road tarmac option -- though perhaps the mountain hiking would be a plus for you ?
The option up the Adour via Tarbes, after the necessary DIY from Tarbes up to Bagnères, would lead to either a rather mountainous option along the Piémont Way to Lourdes, or to a still mountainous but less so multiple-choice DIY possibility on tarmac to Lourdes via various villages.
The GR101 is the "easy" option, including in its possibilities for tarmac variants via towns and villages.
I might personally go the Tarbes Way then DIY out of there along a cyclists path through Juillan & Louey, through Adé to Lourdes. Which I suppose technically constitutes a fourth waymarked possibility. It's just that Tarbes is actually a fairly nice city, and they have good food and etc. It's quite possible BTW that this would be closest to the "historic" route from Maubourguet & Tarbes to Lourdes, and it would certainly seem the more "sensible" one, all waymarks & "typical" hiker stuff excluded.
The GR101 appears to go out of its way to get people into hills and mountains and away from villages and pleasantness for what I would personally consider as no good pilgrim purpose -- though some parallel tarmac routes into the villages that it tries to avoid could be worth it. But I simply can see no advantage in avoiding Tarbes so extremely as the GR101 does, nor even at all !!