The name Bharat is specifically ALSO mentioned in Manimekhalai:

 
Padalsar sirappir Bharadhat tongiya (17-57).

Much sung Bharat was famous for maintaining good relations with other
countriesbor with all weaths to be attracted by other countries.

porkottu imayamum podiyum ponre (Puram,2: 25)
Murinjiyur Naganar praises Cheraman Peunchotru Udhiyan Cheraladhan
that he should live long like the golden peaked Himalayas and
podiyam / Podigai Hills.

Vadathu panipadu neduvarai vadakkum
Thenadhu urukezhu kumariyin terkum
Kunadhu karaiporu todukadal kunakkum
Kudathu tonrumudhir bauvattin kudakkum (Puram.6: 1-4).

Karikizhardescribes the land with the biundaries as follows
1. snow clad tall mountains in the north
2. the fearful Kumari in the south
3. shore washing ocean in the east
4. the old ocean on the west

Tenkumari vadaperungal
Kunakuda kadala ellai (Puram,17: 1-2).
Kurungozhiyur Kizhar describes the boundaries of the land as Kumari
in the south, a big rocky (hill) in the north, eastern ocean in the
east and western ocean in the west.

Kumariam peruturai ayirai idaiyadhu manthi
Vadamalai peyarguvin ayin idaiyadhu
Chozha nanndau (Puram. 67: 6-8)
Pisirandaiyar mentions that Chozhanadu was in between Kumari (in the
south) and the Northern mountain (Himalayas).

Vadathisai yadhuve vantoi imayam
Tenthisai yadhuve Aykudi inrayin (Puram.132: 7-8).
If the sky touching Himalayas in the north and theclan of Ay in the
South were not there, the vast world would get destroyed.

pani bauvattuk
kunakuda kadalodu ayidai (Padit.51:14-15)
Kings and the learned coming from different parts of the land, which
is surrounded by conch sounding cold southern ocean, eastern ocean
and western ocean.

Tenkumari vadaperungal
Kunakudakadala ellai (Maduraikkanchi.70-71)
Mangudi Marudhanar describes the boundaries of the land are the
southern Kumari, northern big mountain, eastern ocean and western
ocean.

Note how Kurungozhiyur Kizhar and Mangudi Marudhanar repeat the
sentences exactly same as if they quote.

Thus, it is evident that the land where the ancient Tamils / Hindus / Bharatiyas lived was known as "Bharat" only.