I could see Sardinia without Tunis if the allies could use the Bizerte airfields.
Sicily without Tunis will have a lot heavier logistics requirements.
Allied logistics for Op Husky & the Sicillian campaign were not based in Tunisia. The naval component did use the estuaries near Bizerte to park the light weight landing craft, but that was a convinence not a necessity & other options are practical. The entire 8t Army was embarked from ports in Lybia & Egypt, & its logistics base was in Alexandria. The US 7th Army was embarked from Algerian & Morrocan ports & the logistics hub was spread across the Algerian ports. 8th Army components, the Canadian Div & some separate supporting units were embarked in the UK and went directly to the invasion beaches. The US 45th ID came from the US, with a 48 hour rembarkation stop in Algeria.
The Axis made a effort to demolish the Bizerte, Tunis, & Sfax port facilities & two months were not enough for the Allies to both repair those ports & establish their main logistics base in Tunisia. As planned they used already established ports/depots.
Allied fighter planes were based in Tunisia to support Op Husky & some of the medium bomber groups. They were also based in Malta & the heavy bomber bases were in Algeria & Lybia. With Sardinia secured in March it is available as a airbase for operations against Sicily & as are its ports for forward bases.
Andersons 1st Army had been training for amphibious ops at least since Op Gymnast had been on the table. Possiblly earlier, but I dont have specific evidence. They had also participated in Op Torch in November. Sardinia at the time was defended by a single Italian corps of two understrength infantry divisions & the usual Italian militia. Its not like the entire 1st Army is going to make a operation Husky or Neptune size assualt. OTL a single ad Hoc French corps was sufficient to secure both islands.
I can't see Tunisia being evacuated in this scenario, and I'm not sure the Allies have the strength to defeat the Axis in Tunisia and fight all the other battles, without more US resources being committed to the Mediterranean.
Defeat of the Axis position in Tunisia does not have to occur through direct ground assualt. The ground battle is best regulated to a holding action. OTL the Axis ground and air forces in Tunisia were defeated through Allied airpower interdicting their supplies. That can occur with Allied airforces based in Algeria as OTL and with the airbases in Sardinia/Corsica allowing the Allied medium bomber groups larger acess to the Italian ports & railways.
The Allies would also have to accept the Mediterranean remaining closed to shipping traffic, and US concerns about the threat through Spain would remain.
Once the Axis air forces were defeated in the central Med, which occured April-May OTL the Allies were able to send ships though the Sicilian straits with acceptable losses (actually nearly no losses). As noted earlier the Brits started cargo ship convoys through the central Med before Sicily was invaded. If it is useful to invade Sicilly then the Sardinian airbase adds the practicality of a north coast invasion while the possesion of Malta keeps open the exposure of the SE coast. This complicates the defense for the Axis as they must prepare for two likely approaches. The Axis defense problem is also complicated if the bulk of their ground forces are still in Tunisia. The OTL defense of Sicilly depended on mobile forces rebuilt from those evacuated from Tunisia. If there is no evacuation the ground forces in Italy are less, and they must cover a larger front as southern Italy can be just as vulnerable with a Allied controlled Sardinia.
Stratigically it makes more sense for the Axis to evacuate Tunisia once Sardinia is in Allied hands. Its a stratigic dead end & extremely vulnerable to isolation. Every battalion there leaves a beach in Sicilly or Italy undefended. However we are limited to Hitlers decisions here, which were not informed by stratigic good sense or solid logic as we understand it. I'd predict the usual Hitler half measure from a desire to have things both ways, with a corps or two removed from Tunisia, but over 200,000 Axis soldiers still at risk of being cut off.
OTL the threat from Spain was finally dismissed. Reinforcements were sent to Tunisia from the US I Armored Corps, & 7th Army planning for follow on Ops like Op Husky removed the remainder of I Corps after March.