Does K2CO3 react in minute amounts with glassware at higher temperatures?
In agriculture, potassium silicate ($\ce{K2SiO3}$) is used as foliar fertilizer ($\ce{K2SiO3}$ is water soluble). The major synthetic method of the preparation of $\ce{K2SiO3}$ is heating $\ce{SiO2}$ and $\ce{K2CO3}$ at high temperatures $(\pu{600-850 ^\circ C})$ in various mole ratios (e.g., Ref.1):
$$\ce{SiO2 + K2CO3 -&gt;[\Delta] K2SiO3 + CO2} \tag1$$
According to Ref.1, alkali metal carbonates such as $\ce{Na2CO3, K2SiO3, Rb2CO3,}$ and $\ce{Cs2CO3}$, all give corresponding metal silicates in this reaction (e.g., silica to metal carbonate mole ratios used in these preparations are $1:2$ to $3:1$).
Keep in mind that the temperatures used in these preparations are below the melting points of both reactants, $\ce{SiO2}$ ($\pu{1713 ^\circ C}$) and $\ce{K2CO3}$ ($\pu{891 ^\circ C}$), thus, the reaction is essentially a solid state one.
Potassium silicate fertilizer grades were successfully produced by direct fusion of silica ($\ce{SiO2}$) and potassium salt ($\ce{KOH}$ and $\ce{K2CO3}$) in furnaces at temperatures up to melting point of the mixtures (melt temperatures are as high as $\pu{1350 ^\circ C}$). The range of the weight ratio of silicon dioxide/potasium solid used in this method is $1:1$ to $5:1$. The reaction between $\ce{SiO2}$ and $\ce{KOH}$ is being:
$$\ce{SiO2 + 2KOH -&gt;[\Delta] K2SiO3 + H2O} \tag2$$
According to the Ref.1, the other salts (e.g., halides and sulfates) of the metal carbonate catalyze the reaction. For example, in preparing sodium silicate, 6.12 parts by weight sodium carbonate, 3.47 parts by weight silica of minus 325 mesh particle size, and 0.41 part by weight sodium sulfate were ground together (which is $\ce{SiO2 + Na2CO3 + 0.05Na2SO4}$). The ground mixture was heated at a temperature of approximately $\pu{700 ^\circ C}$. for $\pu{4.5 h}$, after which time, the reaction was found to be 51% complete (c.f., without catalyst under same condition, completion was ~$23\%$).
Based on these date, it is safe to say that when silica made contact with an alkali carbonate or hydroxide at elevated temperatures, we can expect a reaction between two chemicals, regardless of they are been in solid form or melt. Once washed, the formed metal silicate would be washed away by water, leaving silica with damaged surface. According to Wikipedia:

The most familiar, and historically the oldest, types of manufactured glass are &quot;silicate glasses&quot; based on the chemical compound silica (silicon dioxide, or quartz), the primary constituent of sand. Soda-lime glass, containing around 70% silica, account for around 90% of manufactured glass.

Therefore, it is no wonder when you applied alkali hydroxide solution (e.g., $\ce{NaOH}$ or $\ce{KOH}$) solution to glass surface for a long time and found the surface become roughed. The alkali carbonated would do the same in higher temperatures.
Note: $\ce{Li2SiO3}$ is only partially soluble in water. However, its $\ce{Li4SiO4}$ version has an advantage of being $\ce{CO2}$ absorber (Ref.3) according to the reaction:
$$\ce{Li4SiO4 + CO2 -&gt;[\Delta] Li2SiO3 + Li2CO3} \tag3$$
Unlike other carbonates, two $\ce{Li2CO3}$ molecules react with one $\ce{SiO32}$ molecule to give $\ce{Li4SiO4}$:
$$\ce{2Li2CO3 + SiO2 -&gt;[\Delta] Li4SiO4 + 2CO2} \tag4$$
References:

Isadore Mockrin, “Production of Silicates,” US Patent 1958, 2,823,098 (PDF).
Srie Muljani, Bambang Wahyudi, Ketut Sumada, Suprihatin, “Potassium silicate foliar fertilizer grade from geothermal sludge and pyrophylite,” MATEC Web of Conferences 2016, 58, 01021 (DOI: 10.1051/matecconf/20165801021) (Proceedings of the 3rd Bali International Seminar on Science &amp; Technology)(PDF).
Xianyao Yan, Yingjie Li, Xiaotong Ma, Jianli Zhao, Zeyan Wang, “Performance of $\ce{Li4SiO4}$ Material for CO2 Capture: A Review,” Int. J. Mol. Sci. 2019, 20, 928 (22 pages) (DOI: 10.3390/ijms20040928).