Can I safely replace flat Lipo battery with a Li-Ion round battery pack on a constantly powered device?
Either of the replacement batteries is probably suitable.
A 3.6V or 3.7V lithium ion / lithium polymer battery  will be at about 4.2V when fully charged. Newer chemistries may allow 4.3V. Charging a 4.3V capable battery to 4.2V max will very slightly reduce stored capacity and do no harm. Charging a 4.2V capable battery to 4.3V can be a very very very bad idea.
Test:   Charge the old battery fully when off and note the maximum voltage reached. This is probably 4.2V but may be 4.3V (newer and less likely).
Look at the datasheets (you do have datasheets available, don't you? :-) ) and note the maximum allowed charge voltage. Again, this is probably 4.2V.
If old and new are 4.2V or new is 4.3V capable then there is a very good chance that either of the replacement batteries are suitable.
If you can supply brands and part numbers for all batteries concerned it will help us to supply a better answer.