See, being a Maadhva, my answer can differ from other people. Since you have requested this answer, I will answer it. Please forgive me if I go wrong anywhere. Umm…if you see, then Lalita Tripurasundari is different and Radha is different. I have heard that Lalita Tripurasundari as per Shaktas is one of the 10 Mahavidyas known as Sri Shodashi devi. One must note that SriLalita Tripurasundari is different from SriLalita sakhi of Srimati Radharaniji. This answer will be based on Vaishnava perspective, especially Madhva siddhanta. I don’t know much about Shaktism so I won’t talk from that lens, plus I am not even a Shaakta.
HOW MIGHT SRILALITA TRIPURASUNDARI BE UNDERSTOOD FROM A MADHVA PERSPECTIVE?
From the Madhva perspective, the answer would be no. We do not place Srimati Radharaniji in a position from which Lalita Tripurasundari would be considered her amsha. SriMadhvacharya mentions that Lord Narayana is the supreme. Plus, Mahalakshmi devi is the second one after SriNarayana. And Lakshmi has three forms namely : Sri, Bhu and Durga via which She is the presiding deity for Satva (Sri), Rajas(Bhu) and Tamas(Durga) Guna. All the roopas are identical with Mahalakshmi devi and are non-different from them. Now, here too there are two cases based on what one believes:
1. SriLalita Tripurasundari is Durga Parameshwari:
If you consider Her as Durgadevi then in this case, She may be understood as Durga devi who is the form of Mahalakshmi devi. Because, in Lakshmi Tantra, Lakshmidevi says :
bhadrakALI tathA bhadrA kALI durgA maheSvarI |
triguNA bhagavatpatnI tathA bhagavatI parA ||
Further, the people address Me as BhadrakALi, BhadrA, kALi, Durga, Maheshvari, triguna and by other names. They also call Me as Bhagavati since I am the consort of Bhagavan.
mahattvAcca mahAmAyA mohanAnmohinI matA |
durgA ca durgamatvena bhaktarakshAvidherapi ||
I am worshipped as MahAmAyA since My mAya/mahatvam (greatness) pervades everywhere. I am saluted as Mohini since I immerse all in moham (delusion). I am called Durga because I am hard to attain and also because I protect My bhaktAs.
अहं श्रीश्च महीं चैव दुर्गां चैव सनातनम् ।
बिभर्मी त्रिभिरूपैस्तु जगदेतच्चराचरम् ॥
I am indeed Śrī, I am Mahī (Bhū/Earth), and I am the eternal Durgā. Assuming these three distinct forms, I sustain and carry this entire universe of moving and unmoving beings.
[Lakshmi Tantra, Pancharaatra aagama]
So, as per this, SriLalita Tripurasundari may be understood as Durga roopi Mahalakshmi’s form. Because, Durgadevi is Vishnumaya Durga, even as per Devi Mahatmya. Lakshmi devi can do creation, sustainance and destruction just by Her eyebrow movement. And all this is possible due to Lord Narayana’s grace.
2. SriLalita Tripura Sundari as Parvatidevi’s roopa:
Like Lakshmi devi has Durga roopa, Umadevi also has a Durga roopa. This form is what we call as Kaushiki or Mahasarasvati. In this form She killed Shumbha and Nishumbha named demons. So, if you believe SriLalita Tripurasundari as Parvati devi’s Durga roopa then, She has the sannidhana of Vishnumaya roopi Mahalakshmi, similar to how Mahasarasvati had.
When the devatas were frustrated by the nuisance caused by Shumbha and Nishumbha, they approached Kailasa and prayed to Vishnumaya Durga. At that time, Parvati devi was going for having a bath where She sees the devatas praying to someone. She asks, ‘Whom are you all praying to?’ At the same time, a devi came out of Her body cell, who was dark skinned, but radiant and held a bell, trident, plough, conch shell, pestle (a cylindrical club), discus, bow and arrows in Her hands. Since, She came from Umadevi’s body cell, She was known as Kaushiki (Kosha means cell). She said, ‘They all are worshipping me.’
[Mahasarasvati roopa : Durga roopa of Umadevi]
Earlier Umadevi had dark skin. Once, Lord Shiva called Her Kaali, due to which She felt insulted and did penance to Lord Brahma and asked for a new body. Lord Brahma blessed Her with a new body, which was of Gaura varna or golden color. Since, She got a golden body, She was known as Gowry/Gauri. Now, this dark body didn’t go in vain. Because, devata body never goes waste. Hence, due to Her shakti, the body remained in Her body. Devatas do have this kind of power. Now, when these devatas were calling for Vishnumaya Durga, this Vishnumaya Durga entered that dark skinned body within Parvati’s body. That body had some sannidhana of Umadevi. So, this roopa is Parvati’s Durga roopa known as Mahasaraswati or Kaushiki or Chandika Durga.
So, in this way SriLalita Tripurasundari too might have the presence of Vishnumaya Durga, if you consider Her as Umadevi’s roopa.
As per Madhva sampradaya if you see, Srimati Radharaniji is not considered as Lakshmi avatara, since SrimadBhagavatam or the Mahabharata too doesn’t mention about Her. Plus, SriMadhvacharya too doesn’t mention about Her neither in the SriMahabharata Tatparya Nirnaya, nor the SriBhagavata Tatparya Nirnaya. But, She can be considered as a Gopi who is favorite of Krishna in Vrindavana with Lakshmidevi’s special sannidhana in Her. Srimati Radharaniji is a great devotee of Krishna in Vrindavana, but we don’t consider Her as equal to Mahalakshmidevi, because the Gopikas are the incarnation of Apsaraa strees who had came on the earth to get the sanga of Krishna. Apsaraas have their niyata pati as Lord Narayana, and they worship Lord Narayana as their husband. So, whenever they have Lakshmi sannidhana, Lord Narayana accepts them.
So, we don’t consider Lalita Tripurasundari as an inferior amsha of Srimati Radharaniji, but we don’t either consider Lalita Tripurasundari as the supreme mother ruling the trinity. The above two cases may hold true (Case 1 : Lalita Tripurasundari as Durga Parameshvari and Case 2 : Lalita Tripurasundari as Umadevi’s one form) only iff you believe in it and as per your belief. I am not claiming any certainty because it is completely conditional as to what you believe.
Hope this helps!
Jay SriRam!
ShriKrishnarpanamastu!