In short, yes you do if you don’t have both parts paper and photocard you could be slapped with a fine. Speak to the DVLA, I had something similar last year so I sent the forms back special delivery next day with a covering letter and prepaid special delivery envelope as well and had my licence back that week.
Realistically it’s unlikely to cause you a problem i’ve probably spent several months driving in france over the past few years and never had so much as a sniff from the boys in blue but if you don’t have one it’ll be sods law….
You could take the international number for the DVLA with you just in case to dry and get them to fax a certificate of entitlement but for the £15 it’ll cost for special delivery both ways to replace your licence it’s not worth it. You can take other bits of paper as mentioned above and try and sort it if you do get stopped but it’s going to be a lot of faff!