If you are creating Vector data, then it's best to keep the data in a geographic coordinate System with WGS84 as the datum.
With today's technology, there is no specific advantage to keep your vector data in a custom projection. 6 decimal points are enough to have meter level accuracy, and you aren't going to need any more. Projection on demand is fast enough in today's world; And you have the advantage of being compatible with most global sources like Satellite Imagery and GPS devices etc.
You shouldn't use a UTM zone, since India is covered by several zones, and big states like AP & MP themselves are covered by several UTM zones.
If you are dealing with any Governemt body, do note that you need to follow the NNRMS standards given in the NNRMS document. While the document clearly states (page 47 section 66.4) that you should store the data in Geographic Coordinate system, all government organizations that I have worked with, store even the vector data in the national or state projections.
Update
In Feb 2016, these coordinate systems were added to the EPSG Database. For example, look at https://epsg.io/7755
Second Update
This projection is now available both in the ArcGIS platform, as well as QGIS, Gdal and other projects which use the updated PROJ library.
If you are digitizing from SOI TopoSheets, It is even more complicated. Firstly, many experts believe that it is not legal to scan and digitize SOI TopoSheets. Secondly these toposheets are in a poly-conic projection, based on the Everest Spheroid & Modified Everest Datum. The coordinates on these maps are shifted quite a bit when compared to coordinates in the WGS84 datum. Hence you need to take care of this datum transformation.
You first need to decide what distortion properties would you like to control for. That is to say, are you interested in preserving area, distance or shape? There is a decision support tool for selecting projections that is quite user friendly and available for free from Oregon State University (Map Projection Selection Tool). Note that you can select the distortion property and a specific area of interest. The tool yields a recommended projection, a central meridian and PROJ.4 code.
Best Answer
I just tested it with OSM roadlayer projected into EPSG:3857 and a pointlayer also in 3857 with QGIS 3.22.1 and encountered no error - just as information.
Please retry it with a new profile: settings->user profiles->new profile and if you want, you can download my testdata (https://files.wheregroup.com/index.php/s/omfMWo5YzbNQNEW)